1. Section
Recently, to better align with the insurance practice, there has been a growing trend among applied probability researchers to consider renewal risk models with main and delayed claims. For instance, a severe car accident often results in not only vehicle damage but also injuries or fatalities. As the severity of the accident increases, the extent of the damage tends to worsen, and passengers are more likely to be seriously injured. Concretely, claims for prompt compensation, such as those for property damage and immediate death, are referred to as main claims, whereas those are processed after an uncertain period of time, such as the injuries, are classified as delayed claims.
The classical unidimensional risk model with delayed claims can be traced back to the work of [Reference Waters and Papatriandafylou28] for a discrete-time case and [Reference Yuen, Guo and Ng33] for a continuous-time case, where both main and delayed claims are modeled by random variables (RVs) with light tails. However, most insurance claims, particularly the most severe, are heavy-tailed; see [Reference Embrechts, Klüppelberg and Mikosch9]. In the presence of heavy-tailed claims, in [Reference Li17] the asymptotic results of randomly weighted sums of RVs were investigated and the results were applied to a claim dependence risk model with constant interest force. Geng et al. [Reference Geng, Liu and Wang12] applied a generalized Kesten inequality to randomly weighted sums of dependent subexponential RVs and used the result in a risk model with delayed claims. For extensions with dependence structures and stochastic returns, one can see [Reference Fu and Li10, Reference Li and Wu18, Reference Yang, Wang and Zhang30].
Notably, the insurance practice indicates some dependence between claims and waiting times, making the independence assumption unrealistic. In light of this, Liu et al. [Reference Liu, Chen and Fu21] employed the dependence structure initially introduced in [Reference Asimit and Jones3] via copulas and further developed in [Reference Asimit and Badescu2] on the main claims and proposed a similar structure for the delayed claims. Most recently, Yuan and Lu [Reference Yuan and Lu32] proposed a time-claim-dependent framework that can describe the dependence among waiting times and claims. They established an asymptotic expansion for the finite-time ruin probability.
On the other hand, bidimensional risk models are capable of providing insurers with thorough insights into various aspects of their business compared with unidimensional models. Jiang et al. [Reference Jiang, Wang, Chen and Xu13] studied a bidimensional time-dependent risk model, where they assumed that the two business lines share a common counting process. Lu and Yuan [Reference Lu and Yuan22] incorporated delayed claims into the bidimensional risk models. The asymptotic estimates for four kinds of ruin probabilities were derived under some dependence structures. For more studies on dependence assumptions in bidimensional risk models, readers can see [Reference Chen, Yang and Jiang5, Reference Li and Yang19, Reference Wang, Yang, Liu and Yang27, Reference Yang and Li29] for the constant interest force and [Reference Chen, Li and Cheng6, Reference Wang, Cheng and Yan25, Reference Yang, Chen and Yuen31] for stochastic returns. For multidimensional risk models, readers can refer to [Reference Cheng, Konstantinides and Wang8, Reference Konstantinides and Li15].
It is noteworthy that much of the research on risk models has predominantly focused on either unidimensional models accounting for delayed claims or multidimensional models ruling out delayed claims. Meanwhile, in the limited existing literature on bidimensional delayed-claim risk models, including those mentioned above, few researchers focus on the time-dependent feature because of mathematical tractability. Drawing inspiration from [Reference Yuan and Lu32], we propose a more general dependence framework by replacing the measurable functions with stochastic processes. Moreover, we change the constant force of interest in the risk model into stochastic returns, making it more in line with the insurance practice. These modifications introduce difficulties regarding the uniformity of asymptotic formulas, which we address by investigating a series of properties of the subexponential class. Subsequently, we explore the uniform asymptotic results of the ruin probabilities of a bidimensional risk model with subexponential claims.
We now we give a detailed description of our risk model. Consider a bidimensional risk model with main and delayed claims in which, for
$i\in\mathbb N_+$
and
$k=1,2$
, the insurer’s ith main claim of the kth business line
$X_{ki}$
occurs at time
$\tau_{ki}$
accompanied with a delayed claim
$Y_{ki}$
occurring at time
$\tau_{ki}+D_{ki}$
, where
$D_{ki}$
denotes an uncertain delay time. Assume that for
$k=1,2$
, claims of sizes
$X_{ki}, i\in\mathbb N_+$
arrive according to a renewal counting process
$N_k(t)=\sup\!\left\{n\ge1\;:\;\tau_{kn}\le t\right\}$
with a finite mean function
$\lambda_k(t)=\textrm{E}N_k(t)=\sum_{i=1}^\infty\textrm{P}(\tau_{ki}\le t)$
. For each
$k=1,2$
, denote the interarrival times for the kth line of business
$\tau_{k1},\tau_{k2}-\tau_{k1},\tau_{k3}-\tau_{k2},\ldots$
by
$\theta_{k1},\theta_{k2},\theta_{k3},\ldots$
, which constitute an independent and identically distributed (i.i.d.) nonnegative random sequence. Here
$\!\left\{{(C_1(t),C_2(t))}^\top,t\ge0\right\}$
describes the bidimensional premium income process. Assume that for each
$k=1,2$
,
$C_k(t)$
is nonnegative, nondecreasing, and independent of the other sources of randomness, satisfying
$C_k(0)=0$
and
$C_k(t){\,\lt\,}\infty$
almost surely (a.s.) for
$0 {\,\lt\,} t {\,\lt\,} \infty$
. The vector of initial reserves is represented by
${(x_1,x_2)}^\top$
. Over and above, the insurer is allowed to invest the surplus in a portfolio of risk-free assets. For
$k=1,2$
, the investment return process of each business line is described by a geometric Lévy process
$\!\left\{\textrm{e}^{R_k(t)},t\ge0\right\}$
. Here for
$k=1,2$
,
$\{R_k(t),t\ge0\}$
is a nonnegative Lévy process, which starts from zero and has independent and stationary increments. For more discussion about Lévy process, see [Reference Applebaum1]. Consequently, the surplus process with stochastic returns on investments up to time
$t\ge0$
can be defined as
\begin{align} \!\left(\begin{array}{c}U_1(t)\\[5pt] U_2(t)\end{array}\right)&=\!\left(\begin{array}{c}x_1\textrm{e}^{R_1(t)}\\[5pt] x_2\textrm{e}^{R_2(t)}\end{array}\right)+\!\left(\begin{array}{c}\int_0^t\textrm{e}^{R_1(t-s)}C_1(\mathrm d s)\\[9pt] \int_0^t\textrm{e}^{R_2(t-s)}C_2(\mathrm d s)\end{array}\right)-\!\left(\begin{array}{c}\sum\limits_{i=1}^{N_1(t)}X_{1i}\mathrm{e}^{R_1(t-\tau_{1i})}\\[9pt] \sum\limits_{i=1}^{N_2(t)}X_{2i}\mathrm{e}^{R_2(t-\tau_{2i})}\end{array}\right)\nonumber\\ & \quad -\!\left(\begin{array}{c}\sum\limits_{i=1}^\infty Y_{1i}\mathrm{e}^{R_1(t-\tau_{1i}-D_{1i})}\mathbb I_{\!\left\{\tau_{1i}+D_{1i}\le t\right\}}\\[9pt] \sum\limits_{i=1}^\infty Y_{2i}\mathrm{e}^{R_2(t-\tau_{2i}-D_{2i})}\mathbb I_{\!\left\{\tau_{2i}+D_{2i}\le t\right\}}\end{array}\right),\quad t\ge0,\end{align}
where
$\mathbb I_A$
denotes the indicator function of a set A.
When investigating the risk model (1), we constrain the scale of time in the set:
$\Lambda_k = \{t \ge 0\;:\;0{\,\lt\,}\lambda_k(t)\le\infty\},\ k=1,2.$
Let
$\underline{t}_k = \inf\{t\ge 0\;:\;\mathbb{P}(\tau_{k1}\le t)\}$
. If
$\mathbb{P}(\tau_{k1} = \underline{t}_k){\,\gt\,}0$
then we have
$\Lambda_k =[\underline{t}_k,\infty]$
; if
$\mathbb{P}(\tau_{k1} = \underline{t}_k)=0$
then we have
$\Lambda_k =(\underline{t}_k,\infty]$
. Moreover, let
$\Lambda = \Lambda_1\cap\Lambda_2$
and
$\Lambda_T=[0,T]\cap\Lambda$
for any
$T\in\Lambda$
.
For a bidimensional risk model, there are multiple ways to identify ruin probabilities. Throughout this paper, the following finite-time ruin probabilities with a finite-horizon t are investigated in particular:
\begin{align*}\psi_{\rm sum}(x_1,x_2;\;t)=\textrm{P}\!\left(\inf\limits_{0\le s\le t}\!\left\{U_1(s)+U_2(s)\right\}{\,\lt\,}0\bigg|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right),\end{align*}
\begin{align*}\psi_{\rm or}(x_1,x_2;\;t)=\textrm{P}\!\left(\inf\limits_{0\le s\le t}\!\left\{U_1(s)\wedge U_2(s)\right\}{\,\lt\,}0\bigg|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right),\end{align*}
\begin{align*}\psi_{\rm and}(x_1,x_2;\;t)=\textrm{P}\!\left(\inf\limits_{0\le s\le t}U_1(s){\,\lt\,}0,\inf\limits_{0\le s\le t}U_2(s){\,\lt\,}0\bigg|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right)\end{align*}
and
\begin{align*}\psi_{\rm sim}(x_1,x_2;\;t)=\textrm{P}\!\left(\inf\limits_{0\le s\le t}\!\left\{U_1(s)\vee U_2(s)\right\}{\,\lt\,}0\bigg|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right).\end{align*}
Here
$\psi_{\rm sum}(x_1,x_2;\;t)$
is the probability that the aggregate surplus process
$\!\left\{U_1(s)+U_2(s)\right\}$
falls below zero within the time horizon [0, t];
$\psi_{\rm or}(x_1,x_2;\;t)$
is the probability that at least one of business lines has negative surplus within [0, t];
$\psi_{\rm and}(x_1,x_2;\;t)$
is the probability that both surplus processes of the two business lines become negative within [0, t], but not necessarily at the same time; and
$\psi_{\rm sim}(x_1,x_2;\;t)$
is the probability that the two lines of business have negative surplus within [0, t] simultaneously. From the definitions of ruin probabilities, the following relationships can be easily derived:
\begin{align}\psi_{\rm or}(x_1,x_2;\;t)\ge\psi_{\rm sum}(x_1,x_2;\;t)\ge\psi_{\rm sim}(x_1,x_2;\;t)\end{align}
and
\begin{align}\psi_{\rm or}(x_1,x_2;\;t)\ge\psi_{\rm and}(x_1,x_2;\;t)\ge\psi_{\rm sim}(x_1,x_2;\;t).\end{align}
See [Reference Lu and Zhang23] for similar definitions of ruin probabilities, among many others.
The rest of this paper consists of four sections. Section 2 introduces some necessary preliminaries and then outlines the main result. In Section 3, a simulation study is conducted to verify the accuracy of the asymptotic estimates obtained. Some useful lemmas and the proof of our main result are given in Section 4.
2. Preliminaries and the main result
2.1. Heavy-tailed distributions and Lévy processes
In the following, the following notation is used for convenience. Hereafter, all limit relationships hold as
$x\to\infty$
unless otherwise stated. For two positive functions
$f(\cdot)$
and
$g(\cdot)$
, we write
$f(x)\lesssim g(x)$
or
$g(x){\,\gtrsim\,}\, f(x)$
if
$\limsup_{x\to\infty}f(x)/g(x)\le1$
, write
$f(x)\sim g(x)$
if both
$f(x)\lesssim g(x)$
and
$f(x){\,\gtrsim\,} g(x)$
, write
$f(x)=o(g(x))$
if
$\lim_{x\to\infty}f(x)/g(x)=0$
, write
$f(x)\asymp g(x)$
if
$0 {\,\lt\,} \liminf_{x\to\infty}f(x)/g(x)\le\limsup_{x\to\infty}f(x)/g(x){\,\lt\,}\infty$
, write
$f(x)\wedge g(x) = \min \{f(x),g(x)\}$
and write
$f(x)\vee g(x) = \max \{f(x),g(x)\}$
.
A RV X or its distribution V is said to be heavy-tailed if
$\textrm{E} \textrm{e}^{sX}=\infty$
for any
$s{\,\gt\,}0$
. Let
$V=1-\overline V$
be a distribution on
$\!\left[0,\infty\right)$
with an infinite upper endpoint, i.e.
$\overline V(x){\,\gt\,}0$
for all
$x\ge0$
. A distribution V supported on
$\!\left[0,\infty\right)$
is said to belong to the class of long-tailed distribution class, denoted by
$V\in\mathcal L$
, if for all
$x\ge0$
, the relation
$\lim\limits_{x\to\infty}\overline V(x-y)/\overline V(x)=1$
,
$y\in(-\infty,\infty)$
holds. Set
\begin{align*}\mathcal H(V)=\!\left\{l(x)\mbox{ on }\!\left[0,\infty\right)\;:\;l(x)\uparrow\infty, \ \frac{l(x)}{x}\downarrow0 \mbox{ and }\overline V(x-l(x))\sim\overline V(x) \right\}.\end{align*}
If
$V\in\mathcal L$
, there exists a function
$l(x)\in\mathcal H(V)$
such that
$\overline V(x-Kl(x))\sim\overline V(x)$
for every
$K{\,\gt\,}0$
, see Lemma 4.1 of [Reference Li, Tang and Wu20]. Furthermore, for any fixed
$n\in \mathbb N _+$
, if
$V_i\in\mathcal L$
and
$l_i(x)\in\mathcal H(V_i)$
for
$1\le i\le n$
, we have
$\overline V_i(x-l(x))\sim\overline V_i(x)$
for all
$1\le i\le n$
, where
$l(x)=\min\limits_{1\le i\le n}l_i(x)$
, and write
$l(x)\in\mathcal H(V_1,\ldots,V_n)$
.
A distribution V belongs to the subexponential class, written as
$V\in\mathcal S$
, if
$\overline{V^{n*}}(x)\sim n\overline V(x)$
holds for all (or, equivalently, for some)
$n\ge2$
, where
$V^{n*}$
is the n-fold convolution of V. The class
$\mathcal S$
contains a lot of important distributions such as Pareto, lognormal, and heavy-tailed Weibull distributions.
For a distribution V, denote
$\overline V^*(y)=\limsup_{x\to\infty}\overline V(xy)/\overline V(x)$
for
$y{\,\gt\,}0$
. A distribution V supported on
$\!\left[0,\infty\right)$
belongs to the dominated variation class, written as
$V\in\mathcal D$
, if for any (or, equivalently, for some)
$y\in(0,1)$
,
$\overline V^*(y){\,\lt\,}\infty$
. In addition,
$\mathcal L\cap\mathcal D$
is a class of heavy-tailed distributions with wide application.
A distribution V is said to be regularly varying tailed with index
$\alpha{\,\gt\,}0$
, denoted by
$V\in\mathcal R_{-\alpha}$
, if
$\overline V(xy)\sim y^{-\alpha}\overline V(x)$
for all
$y{\,\gt\,}0$
. The heavy-tailed distributions satisfy the following relation:
$\mathcal R_{-\alpha}\subset\mathcal L\cap\mathcal D\subset\mathcal S\subset\mathcal L$
. For more details on the heavy-tailed distributions, readers can see [Reference Embrechts, Klüppelberg and Mikosch9].
Assume for
$k=1,2$
, the Lévy process
$\!\left\{R_k(t),t\ge0\right\}$
has Lévy triplet
$(\gamma_k,\sigma_k^2,\rho_k)$
, where
$-\infty{\,\lt\,}\gamma_k{\,\lt\,}\infty$
and
$\sigma_k{\,\gt\,}0$
are two constants, and
$\rho_k$
is a Lévy measure on
$(-\infty,\infty)$
satisfying
$\rho_k(\{0\})=0$
and
$\int_{-\infty}^\infty(x^2\wedge1)\rho_k(\mathrm d x){\,\lt\,}\infty$
. For
$k=1,2$
, the Laplace exponent of a Lévy process is defined as
$\phi_k(z)=\log\textrm{E} \mathrm{e}^{-zR_k(1)},\ z\in\mathbb R.$
If
$\phi_k(z)$
is finite then for any
$t\ge 0$
,
$\textrm{E}\mathrm{e}^{-zR_k(t)}=\mathrm{e}^{t\phi_k(z)}{\,\lt\,}\infty$
and we have that
$\phi_k(z)$
is convex in z.
2.2. Dependence structures and verification
Practically, a multitude of dependence structures find application in modeling dependence among claims and interarrival times. Among many others, Yuan and Lu [Reference Yuan and Lu32] introduced the following time-dependence assumption which is based on the idea of [Reference Asimit and Badescu2].
Assumption 2.1. For
$k=1,2$
, there exists a univariate measurable function
$h_k(\cdot)\;:\left [0,\infty\right)\rightarrow(0,\infty)$
and a bivariate measurable function
$\varphi_k(\cdot,\cdot)\;:\;{\left [0,\infty\right)}^2\rightarrow(0,\infty)$
satisfying
\begin{align}0{\,\lt\,}\inf\limits_{s\in\Lambda_T}h_k(s)\le\sup\limits_{s\in\Lambda_T}h_k(s){\,\lt\,}\infty\end{align}
and
\begin{align}0{\,\lt\,}\inf\limits_{s,u\in\Lambda_T}\varphi_k(s,u)\le\sup\limits_{s,u\in\Lambda_T}\varphi_k(s,u){\,\lt\,}\infty\end{align}
such that the random vector
$(X_k,Y_k,\theta_k,D_k)$
fulfills the relations
\begin{align}\textrm{P}\!\left(X_k{\,\gt\,}x \mid\theta_k=s\right)\sim \textrm{P}(X_k{\,\gt\,}x)h_k(s)\end{align}
and
\begin{align}\textrm{P}\!\left(Y_k{\,\gt\,}x \mid\theta_k=s,D_k=u\right)\sim \textrm{P}(Y_k{\,\gt\,}x)\varphi_k(s,u)\end{align}
uniformly for
$s,u\in \Lambda_T$
.
Remark 2.1. When s and u are not a possible value of
$\theta$
and D, the conditional probabilities shown in (6) and (7) are simply understood as unconditional probabilities so that
$h(s)=1$
and
$\varphi(s,u)=1$
. Here the uniformity of (6) and (7) is understood as
\begin{align*}\lim\limits_{x\to\infty}\sup\limits_{s\in\Lambda_T}\!\left|\frac{\textrm{P}\!\left(X_k{\,\gt\,}x|\theta_k=s\right)}{\textrm{P}\!\left(X_k{\,\gt\,}x\right)h_k(s)}-1\right| = 0\end{align*}
and
\begin{align*}\lim\limits_{x\to\infty}\sup\limits_{s,u\in\Lambda_T}\!\left|\frac{\textrm{P}\!\left(Y_k{\,\gt\,}x|\theta_k=s,D_k=u\right)}{\textrm{P}\!\left(Y_k{\,\gt\,}x\right)\phi_k(s,u)}-1\right| = 0.\end{align*}
This time-dependent structure adapts some commonly used bivariate copulas such as the Farlie–Gumbel–Morgenstern (FGM) copula, the Ali–Mikhail–Haq copula, and the Frank copula. See [Reference Asimit and Badescu2, Reference Li, Tang and Wu20] for a detailed discussion on (6) and [Reference Yuan and Lu32] on (7).
The following assumption defines a general dependence structure capturing the dependence among the main claim, the delayed claim and their corresponding waiting time, which is a natural generalization of (2.4) in [Reference Yuan and Lu32]. It provides increased flexibility for coping insurance practice since the positive measurable functions are replaced by geometric Lévy processes.
Assumption 2.2. There are some nonnegative Lévy processes
$\!\left\{R(t),t\ge0\right\}$
independent of
$(X,Y,\theta,D)$
such that the relation
\begin{align}&\textrm{P}\!\left(X\mathrm{e}^{-R(\theta)} + Y\mathrm{e}^{-R(\theta+D)} {\,\gt\,} x \big|\theta=s,D=u \right)\nonumber\\[5pt] &\sim \textrm{P}\Big(X\mathrm{e}^{-R(s)} {\,\gt\,} x\Big)h(s)+\textrm{P}\Big(Y\mathrm{e}^{-R(s+u)} {\,\gt\,} x\Big)\varphi(s,u)\end{align}
holds uniformly for
$s,u\in \Lambda_T$
.
Remark 2.2. We point out that in [Reference Yuan and Lu32], the terms
$g_1(\theta)$
and
$g_2(\theta,D)$
on the right-hand side of (2.4) should be written as
$g_1(s)$
and
$g_2(s,u)$
. As a generalization of that relation, we have corrected this mistake accordingly in (8).
We use (8) to describe the dependence within
$\!\left(X,Y,\theta,D\right)$
, which appears to contain some uncertain relations of X and Y. Similarly as discussed in [Reference Yuan and Lu32], we first characterize the tail probability of
$X\textrm{e}^{-R(\theta)} + Y\textrm{e}^{-R(\theta+D)}$
. When X, Y,
$\theta$
, and D are mutually independent, with distributions F,
$G\in\mathcal L$
, and
$\overline F(x)\asymp\overline G(x)\asymp\overline V(x)$
for some
$V\in\mathcal S$
, it follows from Theorem 1 of [Reference Tang and Yuan24] that (8) holds. Moreover, Yuan and Lu [Reference Yuan and Lu32] proposed a conditional of the pairwise strongly quasi-asymptotic independence (PSQAI):
\begin{align}&\lim\limits_{x\wedge y\to\infty}\sup\limits_{s,u\in\Lambda_T}\textrm{P}\!\left(Y{\,\gt\,}y|X{\,\gt\,}x,\theta=s,D=u\right)\nonumber\\[5pt] &\quad =\lim\limits_{x\wedge y\to\infty}\sup\limits_{s,u\in\Lambda_T}\textrm{P}\!\left(X {\,\gt\,} x|Y {\,\gt\,} y,\theta=s,D=u\right)=0\end{align}
as a specific example satisfying the dependence relation, see Lemma A.5 of [Reference Yuan and Lu32]. We find that relation (2.6) remains a sufficient condition for Assumption 2.2, whose proof is presented in Lemma 4.5. In addition, (8) holds uniformly for different form of R(t) provided that
$0\le R(t)\le b$
a.s., where
$b{\,\lt\,}\infty$
is a constant. It can be proved that some commonly used copulas, such as the FGM copula and the Frank copula, satisfy relation (9), see Section 3.2 of [Reference Yuan and Lu32] for a detailed verification of this.
2.3. Main result
Consider the bidimensional risk model (1). Hereafter, we assume that for
$k=1,2$
,
$\{(X_{ki},Y_{ki},\theta_{ki},D_{ki}),i\in\mathbb N_+\}$
is a sequence of i.i.d. copies of a generic vector
$(X_k,Y_k,\theta_k,D_k)$
with generic marginal distributions
$F_k$
,
$G_k$
,
$H_{\theta_k}$
, and
$H_{D_k}$
, respectively. Let
$D_k$
be independent of
$(X_k,\theta_k)$
. Assume that
$\{(X_{1i},Y_{1i},\theta_{1i},D_{1i}), i\in\mathbb N_+\}$
and
$\{(X_{2i},Y_{2i},\theta_{2i},D_{2i}),i\in\mathbb N_+\}$
are mutually independent. For
$k=1,2$
, the Lévy process
$\{R_k(t),t\ge0\}$
is independent of the other sources of randomness and its Laplace exponent
$\phi_k(z)$
is finite. Finally, allow that the premium income process
$C_1(t)$
and
$C_2(t)$
are arbitrarily dependent. We are now ready to state the main result, regarding explicit asymptotic expansions for the four ruin probabilities shown in Section 1.
Theorem 2.1. Consider the bidimensional risk model (1) with
$F_k,G_k\in\mathcal S$
,
$k=1,2$
, and
$\overline F_i(x)\asymp\overline G_j(x)$
,
$i,j=1,2$
. Assume that there exist some functions
$h_k(\cdot)$
and
$\varphi_k(\cdot,\cdot)$
such that
$(X_k, Y_k,\theta_k, D_k)$
satisfies Assumption 2.1 for
$k=1,2$
. Moreover, assume that
$R_k(t)$
makes
$(X_k, Y_k,\theta_k, D_k)$
satisfy Assumption 2.2. For
$k=1,2$
,
$\textrm{P}\!\left(\int_0^T\textrm{e}^{-R_k(s)}C_k(\mathrm d s){\,\gt\,}x\right)=o(\overline F_k(x/a))$
for any fixed
$T\in\Lambda$
and some
$a{\,\gt\,}0$
. Then as
$x_1 \wedge x_2\to\infty$
, we have
\begin{align}\psi_{\rm sum}(x_1,x_2;\;t)&\sim \sum\limits_{k=1}^2\!\left(\int_{0-}^t \textrm{P}\!\left(X_k\textrm{e}^{-R_k(s)}{\,\gt\,} x_1+ x_2\right) {\tilde\lambda}_k(\mathrm d s)\right.\nonumber\\[8pt] &\quad+ \left.\int_{0-}^t \int_{0-}^{t-u}\textrm{P} \!\left( Y_k \textrm{e}^{-R_k(s+u)} {\,\gt\,} x_1+ x_2\right) {\hat\lambda}_k(\mathrm d s,u) H_{D_k }(\mathrm d u)\right),\end{align}
\begin{align}& \psi_{\rm or}(x_1,x_2;\;t)\nonumber\\[3pt] & \;\sim\sum\limits_{k=1}^2\!\left(\int_{0-}^t \textrm{P} \!\left( X_k\textrm{e}^{-R_k(s)}{\,\gt\,} x_k \right) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t \int_{0-}^{t-u} \textrm{P} \!\left( Y_k\textrm{e}^{-R_k(s+u)} {\,\gt\,} x_k \right) {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\end{align}
and
\begin{align}&\psi_{\rm and}(x_1,x_2;\;t)\sim\psi_{\rm sim}(x_1,x_2;\;t)\nonumber\\[5pt] & \quad \sim \prod\limits_{k=1}^2\!\left(\int_{0-}^t\textrm{P}\!\left( X_k\textrm{e}^{-R_k(s)}{\,\gt\,} x_k\right) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t \int_{0-}^{t-u} \textrm{P} \!\left( Y_k\textrm{e}^{-R_k(s+u)} {\,\gt\,} x_k \right) {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u )\right)\end{align}
holds uniformly for
$t\in\Lambda_T$
, where
${\tilde\lambda}_k(t)=\int_{0-}^t (1+\lambda_k(t-s))h_k(s)H_{\theta_k}(\mathrm d s)$
and
${\hat\lambda}_k(s,u)=\int_{0-}^s(1+\lambda_k(s-v))\varphi_k(v,u)H_{\theta_k}(\mathrm d v)$
.
Note that Theorem 2.1 and Lemmas 4.2–4.4 in Section 4 can be extended to the multidimensional case by substituting 2 with d, provided that the RVs from each business line, including the claim sizes, the interarrival times, and the return process, are independent across different lines. Given the analogous method of the proof and its complexity, we omit a detailed presentation here.
The following corollary presents a simplified version of Theorem 2.1, restricting claim distributions to the class of
$\mathcal R_{-\alpha}$
. This result follows through direct application of Breiman’s lemma combined with the definition of the Laplace exponent.
Corollary 2.1. Under the conditions of Theorem 2.1, assume that
$F_k,G_k\in\mathcal R_{-\alpha}$
for some
$0 {\,\lt\,} \alpha {\,\lt\,} \infty$
and let the Laplace exponent of
$\{R_k(t);\;t\ge 0\}$
satisfy
$\phi_k(\beta){\,\lt\,}\infty$
for some
$\beta{\,\gt\,}\alpha$
. Then as
$x_1 \wedge x_2\to\infty$
, it holds uniformly for
$t\in\Lambda_T$
that
\begin{align}\psi_{\rm sum}(x_1,x_2;\;t)\sim\sum\limits_{k=1}^2 &\!\left[\!\left(\int_{0-}^t \textrm{e}^{s\phi_k(\alpha)} {\tilde\lambda}_k(\mathrm d s)\right){\overline F}_k \!\left(x_1+x_2\right)\right.\nonumber\\[5pt] &\!\left.+\!\left(\int_{0-}^t\int_{0-}^{t-u} \textrm{e}^{(s+u)\phi_k(\alpha)}{\hat\lambda}_k(\mathrm d s,u) H_{ D_k }(\mathrm d u)\right)\overline G_k \!\left( x_1+x_2\right)\right],\end{align}
\begin{align}&\psi_{\rm or}(x_1,x_2;\;t)\nonumber\\[5pt] &\sim \sum\limits_{k=1}^2\!\left[\!\left(\int_{0-}^t \textrm{e}^{s\phi_k(\alpha)} {\tilde\lambda}_k(\mathrm d s)\right){\overline F}_k \!\left(x_k\right)+\!\left(\int_{0-}^t \int_{0-}^{t-u} \textrm{e}^{(s+u)\phi_k(\alpha)} {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\overline G_k \!\left( x_k \right)\right]\end{align}
and
\begin{align}&\psi_{\rm and}(x_1,x_2;\;t)\sim\psi_{\rm sim}(x_1,x_2;\;t)\nonumber\\[5pt] &\sim \prod\limits_{k=1}^2\!\left[\!\left(\int_{0-}^t \textrm{e}^{s\phi_k(\alpha)} {\tilde\lambda}_k(\mathrm d s)\right){\overline F}_k \!\left(x_k \right)+\!\left(\int_{0-}^t \int_{0-}^{t-u} \textrm{e}^{(s+u)\phi_k(\alpha)} {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\overline G_k \!\left(x_k\right)\right],\end{align}
where
${\tilde\lambda}_k(t)=\int_{0-}^t (1+\lambda_k(t-s))h_k(s)H_{\theta_k}(\mathrm d s)$
and
${\hat\lambda}_k(s,u)=\int_{0-}^s(1+\lambda_k(s-v))\varphi_k(v,u)H_{\theta_k}(\mathrm d v).$
3. Numerical studies
In this section, we examine the accuracy of asymptotic estimates obtained in Theorem 2.1. We compare the four simulated ruin probabilities with the corresponding asymptotic results via the crude Monte Carlo method. The basic model specifications are listed in the following. The main claims
$X_{1}$
and
$X_{2}$
follow a common Pareto distribution
\begin{align*}F(x)=1-{\!\left(\frac{\gamma}{x+\gamma}\right)}^\alpha,\quad x{\,\gt\,}0,\end{align*}
where the shape parameter
$\alpha=1.5$
and the scale parameter
$\gamma=1$
. Similarly, let the delayed claims
$Y_{1}$
and
$Y_{2}$
follow a common Pareto distribution with
$\alpha=1.5$
and
$\gamma=2$
. The interarrival times of main claims
$\theta_1$
and
$\theta_2$
are exponentially distributed with rate 1 such that the renewal counting processes
$N_1(t)$
and
$N_2(t)$
are two homogeneous Poisson processes both with intensity
$\lambda=1$
. The uncertain delayed times
$D_1$
and
$D_2$
are i.i.d. with a common exponential distribution with rate 1. Moreover, the dependence structure of
$(X_k,Y_k,\theta_k,D_k)$
is modeled by the FGM copula
\begin{align*}\hat{C}(u_1,u_2,u_3,u_4)=u_1u_2u_3u_4\!\left[1+\sum\limits_{l=2}^4\sum\limits_{1\le j_1 {\,\lt\,} \cdots {\,\lt\,} j_l\le4}\vartheta_{j_1\cdots j_l}\prod\limits_{k=1}^l(1-u_{j_k})\right],\end{align*}
where the parameters satisfy the following constraint:
\begin{align*}1+\sum\limits_{l=2}^4\sum\limits_{1\le j_1{\,\lt\,}\cdots{\,\lt\,}j_l\le4}\varepsilon_{j_1}\cdots\varepsilon_{j_l}\vartheta_{j_1\cdots j_l}{\,\gt\,}0, \quad \varepsilon_1=\pm1,\ldots,\varepsilon_4=\pm1.\end{align*}
In the simulation studies, according to the discussion of [Reference Yuan and Lu32], we set
$\vartheta_{14}$
,
$\vartheta_{34}$
, and
$\vartheta_{134}$
to 0 to maintain the independence between D and
$(X,\theta)$
, whereas the remaining parameters are all 0.1. The stochastic return processes
$\{R_k(t),t\ge0\}$
are specialized to
\begin{align*}R_k(t)=0.03t+\sum\limits_{i=1}^{M_k(t)}Z_{ki},\end{align*}
where
$M_k(t)$
is a homogeneous Poisson process with intensity
$\rho_k=1$
for
$k=1,2$
and
$\{Z_{ki},i\ge1\}$
are i.i.d. nonnegative RVs uniformly distributed on interval [0,1], independent of
$M_k(t)$
. According to Proposition 3.10 of [Reference Embrechts, Klüppelberg and Mikosch9], it is easily shown that such an
$R_k(t)$
constitutes a nonnegative Lévy process with
\begin{align*}\phi_k(z)=\log\textrm{E}\textrm{e}^{-zR_k(1)}=-0.03z+\!\left(\textrm{E} \textrm{e}^{-zZ_{k1}}-1\right)\rho_k.\end{align*}
Assume the premium accumulation processes
$C_1(t)=C_2(t)=ct$
for
$t\ge0$
, where the constant premium rate is specified
$c=9.85{\,\gt\,}\textrm{E}X_1+\textrm{E}Y_1$
. Assume that the initial reserve x is equally allocated to the two business lines, that is
$x_1=x_2$
. Finally, the finite time scale
$T=100$
. Consequently, according to the definitions of ruin in Section 1, the ruin probabilities can be estimated by
\begin{align*}\hat\psi_{\rm sum}(x_1,x_2;\;T)=\frac{1}{m}\sum\limits_{j=1}^{m}\mathbb I_{\!\left\{\min\limits_{k=1,\ldots,n}(U_1(t_k)^{(j)}+ U_2(t_k)^{(j)}){\,\lt\,}0\right\}},\end{align*}
\begin{align*}\hat\psi_{\rm or}(x_1,x_2;\;T)=\frac{1}{m}\sum\limits_{j=1}^{m}\mathbb I_{\!\left\{\min\limits_{k=1,\ldots,n}(U_1(t_k)^{(j)}\wedge U_2(t_k)^{(j)}){\,\lt\,}0\right\}},\end{align*}
\begin{align*}\hat\psi_{\rm and}(x_1,x_2;\;T)=\frac{1}{m}\sum\limits_{j=1}^{m}\mathbb I_{\!\left\{\min\limits_{k=1,\ldots,n}U_1(t_k)^{(j)}{\,\lt\,}0, \min\limits_{k=1,\ldots,n}U_2(t_k)^{(j)}{\,\lt\,}0\right\}}\end{align*}
and
\begin{align*}\hat\psi_{\rm sim}(x_1,x_2;\;T)=\frac{1}{m}\sum\limits_{j=1}^{m}\mathbb I_{\!\left\{\min\limits_{k=1,\ldots,n}(U_1(t_k)^{(j)}\vee U_2(t_k)^{(j)}){\,\lt\,}0\right\}},\end{align*}
where
$U_i(t_k)^{(j)}$
is the surplus of each business line at time
$t_k$
for path j.
We conduct
$m=10^7$
simulations to estimate
$\psi_{\rm sum}$
,
$\psi_{\rm or}$
,
$\psi_{\rm and}$
, and
$\psi_{\rm sim}$
. Figure 1 shows the simulated values of the four types of ruin probabilities in Section 1, which verifies relations (2) and (3). As is seen, the probabilities
$\psi_{\rm and}$
and
$\psi_{\rm sim}$
are too small compared with
$\psi_{\rm or}$
, which is slightly bigger than
$\psi_{\rm sum}$
. Further, given that the values of
$\psi_{\rm and}$
and
$\psi_{\rm sim}$
are too small, we continue to compare the simulated values of
$\psi_{\rm or}$
and
$\psi_{\rm sum}$
with the asymptotic formulas in Theorem 2.1, respectively. Figure 2 presents that the ratios stay around 0 and the errors are less than 5%.
Comparison of four types of ruin probabilities.
Figure 1 Long description
A line graph compares four types of ruin probabilities in a classical unidimensional risk model with delayed claims. The x-axis represents the sum of initial reserves of two business lines, ranging from 2000 to 4000. The y-axis represents the simulations of the ruin probabilities, ranging from 0.00000 to 0.00015. Four data lines are plotted: one in orange labeled as Psi or, one in green labeled as Psi sum, one in red labeled as Psi and, and one in blue labeled as Psi sim. The Psi sum line starts at approximately 0.00015 and decreases steadily to around 0.00005. The Psi or line starts at approximately 0.00005 and decreases gradually to around 0.00002. The Psi and and Psi sim lines remain constant at 0.00000 throughout the range. All values are approximated.
Comparison of simulation and approximation of ruin probability (a)
$\psi_{\text{or}}$
and (b)
$\psi_{\text{sum}}$
.
Figure 2 Long description
The image contains two line graphs labeled (a) and (b). Each graph plots the difference between simulations and approximations of ruin probability against the sum of initial reserves of two business lines. Graph (a) shows data for the metric ψ_or, while graph (b) shows data for the metric ψ_sum. Both graphs have the x-axis labeled 'The sum of initial reserves of two business lines' and the y-axis labeled 'Simulations/Approximation-1'. The y-axis ranges from -0.10 to 0.10. The line in graph (a) fluctuates with peaks and troughs, indicating variations in the ruin probability for ψ_or. The line in graph (b) also shows fluctuations but with a different pattern, reflecting variations in the ruin probability for ψ_sum. All values are approximated.
4. Proofs
4.1. Some useful lemmas
Before proving our main result, we prepare some technical lemmas. In addition to their critical role in proving Theorem 2.1, some of these lemmas are themselves interesting.
Lemma 4.1. Assume that X is a RV with the distribution
$F\in\mathcal L$
. Let
$\!\left\{L(t),t\ge0\right\}$
be a nonnegative and nonincreasing stochastic process and independent of X. In addition, assume that L(t) satisfies
$\inf\limits_{t\in\Lambda_T}L(t){\,\gt\,}0$
a.s. and
$\textrm{P}(0 {\,\lt\,} L(t)\le b)=1$
for some
$0 {\,\lt\,} b {\,\lt\,} \infty$
. Denote by
$W_{t\cdot}$
the distribution of XL(t). Then we have
$W_{t\cdot}\in\mathcal L$
for any fixed
$t\in\Lambda_T$
. In addition, for any
$l(x)\in\mathcal H(F)$
, it can be deduced that
$l(x)\in\mathcal H(W_{t\cdot})$
and
\begin{align}\textrm{P}\!\left(XL(t){\,\gt\,}x-l(x)\right)\sim\textrm{P}\!\left(XL(t){\,\gt\,}x\right)\end{align}
holds uniformly for
$t\in\Lambda_T$
.
Proof. The inclusion of
$W_{t\cdot}\in\mathcal L$
and
$\mathcal H(F)\subset\mathcal H(W_{t\cdot})$
for any
$t\in\Lambda_T$
directly follows from Lemma 2.2 of [Reference Geng, Ji and Wang11], which simplifies our objective to proving (16). To demonstrate this, it suffices to show
\begin{align}\lim\limits_{x\to\infty}\sup\limits_{t\in\Lambda_T}\frac{\textrm{P}\!\left(XL(t){\,\gt\,}x-l(x)\right)}{\textrm{P}\!\left(XL(t){\,\gt\,}x\right)}\le 1.\end{align}
Moreover, note that
$\inf_{t\in\Lambda_T}L(t){\,\gt\,}0$
a.s., thus there exists some constant
$t_0\in(0,b)$
such that
$\textrm{P}(L(t){\,\gt\,}t_0){\,\gt\,}0$
. For any
$\varepsilon{\,\lt\,}\min\{t_0,1\}$
, we have
\begin{align}\textrm{P}\!\left(XL(t){\,\gt\,}x-l(x)\right)&=\textrm{P}\!\left(XL(t){\,\gt\,}x-l(x),L(t){\,\gt\,}\varepsilon\right)+\textrm{P}\!\left(XL(t){\,\gt\,}x-l(x),L(t)\le\varepsilon\right)\nonumber\\[5pt] &\;:\!=\;P_1(x,t)+P_2(x,t).\end{align}
For
$P_1(x,t)$
, utilizing Lemma 3.1 of [Reference Cheng and Cheng7] in the second step derives that
\begin{align}P_1(x,t)&=\int_{\varepsilon}^b \textrm{P}(Xu{\,\gt\,}x-l(x))\textrm{P}\!\left(L(t)\in\textrm{d} u\right)\nonumber\\[2pt] &\sim \int_{\varepsilon}^b \textrm{P}(Xu{\,\gt\,}x)\textrm{P}\!\left(L(t)\in\textrm{d} u\right)\nonumber\\[2pt] &=\textrm{P}\!\left(XL(t){\,\gt\,}x,L(t){\,\gt\,}\varepsilon\right)\nonumber\\[2pt] &\le\textrm{P}\!\left(XL(t){\,\gt\,}x\right).\end{align}
On the other hand, noting that
$\!\left\{L(t),t\ge0\right\}$
is nonnegative and nonincreasing, and using
$l(x)\uparrow\infty$
and
${l(x)}/{x}\downarrow0$
in the second step, we get
\begin{align}P_2(x,t)&\le\textrm{P}\!\left(X\varepsilon{\,\gt\,}x-l(x),L(t)\le\varepsilon\right)\nonumber \\[2pt] &\le\textrm{P}\!\left(X\varepsilon{\,\gt\,}x-l\!\left(\frac{x}{\varepsilon}\right)\right)\textrm{P}\!\left(L(t)\le\varepsilon\right)\nonumber\\[2pt] &\sim\textrm{P}\!\left(X\varepsilon{\,\gt\,}x\right)\textrm{P}\!\left(L(t)\le\varepsilon\right)\nonumber\\[2pt] &\le\frac{\textrm{P}\!\left(XL(t){\,\gt\,}x\right)\textrm{P}\!\left(L(t)\le\varepsilon\right)}{\textrm{P}\!\left(L(t){\,\gt\,}\varepsilon\right)}\nonumber\\[2pt] &\le\textrm{P}\!\left(XL(t){\,\gt\,}x\right)\frac{\textrm{P}\!\left(L(T)\le\varepsilon\right)}{\textrm{P}\!\left(L(T){\,\gt\,}\varepsilon\right)},\end{align}
where the third step is attributed to Lemma 4.1 of [Reference Li, Tang and Wu20] while in the penultimate step we use the fact that
$\textrm{P}\!\left(X\varepsilon{\,\gt\,}x,L(t){\,\gt\,}\varepsilon\right)\le\textrm{P}\!\left(XL(t){\,\gt\,}x\right)$
alongside the independence of L(t) and X. Substituting (19) and (20) into (18) and letting
$\varepsilon\downarrow0$
yields the desired relation (17). This completes the proof.
The following lemma investigates the asymptotic tail behavior of the partial sum of aggregate claims from two business lines conditioning on the waiting time, which is critical to the proof of Lemma 4.4. Hereafter, we write
$t_{kn}=\sum\limits_{i=1}^ns_{ki}$
,
$k=1,2$
.
Lemma 4.2. Consider the bidimensional risk model (1) with
$F_k,G_k\in\mathcal L$
,
$k=1,2$
and
$\overline F_i\asymp\overline G_j\asymp \overline F$
,
$i,j=1,2$
for some
$\overline F\in\mathcal S$
. Assume that there exist some functions
$h_k(\cdot)$
and
$\varphi_k(\cdot,\cdot)$
such that
$(X_k, Y_k,\theta_k, D_k)$
satisfies Assumption 2.1 for
$k=1,2$
. Moreover, assume that
$R_k(t)$
makes
$(X_k, Y_k,\theta_k, D_k)$
satisfy Assumption 2.2 for
$k=1,2$
. Then we have for every
$n_1,n_2\in\mathbb N_+$
, it holds uniformly for
$s_{1i}, u_{1i}, s_{2j}, u_{2j}\in\Lambda_T$
,
$1\le i\le n_1$
,
$1\le j\le n_2$
, and
$t\in\Lambda_T$
that
\begin{align}&\textrm{P}\bigg(\!\sum\limits_{k=1}^2\!\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+ Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\{\tau_{ki}+D_{ki}\le t\}}{\,\gt\,} x\bigg| \theta_{ki}=s_{ki},D_{ki}=u_{ki},1\le i \le n_k,k = 1,\!2\!\Bigg)\nonumber\\[2pt] &\quad \sim \sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} \bigg(\textrm{P}\!\left(X_{ki}\textrm{e}^{-R_k(t_{ki})}{\,\gt\,} x\Big|\theta_{ki}=s_{ki}\right)\nonumber\\[2pt] &\quad\quad+\textrm{P}\!\left(Y_{ki}\textrm{e}^{-R_k(t_{ki}+u_{ki})}\mathbb I_{\{t_{ki}+u_{ki}\le t\}}{\,\gt\,} x\Big|\theta_{ki}=s_{ki},D_{ki}=u_{ki}\right)\bigg).\end{align}
Proof. Considering notational simplicity, for
$k=1,2$
, we write
\begin{align*}Z_{ki}=X_{ki}\textrm{e}^{-R_k(t_{ki})}+ Y_{ki}\textrm{e}^{-R_k(t_{ki}+u_{ki})}\mathbb I_{\{t_{ki}+u_{ki}\le t\}}.\end{align*}
To begin with, we shall validate that
\begin{align}&\textrm{P}(Z_{11}+Z_{21}{\,\gt\,}x\mid\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2)\nonumber\\[2pt] &\sim \sum\limits_{k=1}^{2} \bigg(\textrm{P}\!\left(X_{k1}\textrm{e}^{-R_k(t_{k1})}{\,\gt\,}x\Big|\theta_{k1}=s_{k1}\right)\nonumber\\[2pt] &\;\;\;\;\;\;+\textrm{P}\!\left(Y_{k1}\textrm{e}^{-R_k(t_{k1}+u_{k1})}\mathbb I_{\{t_{k1}+u_{k1}\le t\}}{\,\gt\,}x\Big|\theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)\bigg)\end{align}
holds uniformly for
$s_{k1},u_{k1},t\in\Lambda_T,\ k=1,2$
. We split the left-hand side of (22) into three parts
$I_1(x,t)$
,
$I_2(x,t)$
, and
$I_3(x,t)$
according to the value of
$Z_{21}$
belonging to
$\!\left(0,l(x)\right]$
,
$\!\left(l(x),x-l(x)\right]$
, and
$(x-l(x),\infty)$
for some
$l(x)\in\mathcal H(F_1,F_2,G_1,G_2)$
.
For
$I_1(x,t)$
, it holds uniformly for
$s_{k1},u_{k1}\in \Lambda_T,k=1,2,$
that
\begin{align}I_1(x,t)&=\textrm{P}\!\left(Z_{11}+Z_{21}{\,\gt\,}x,Z_{21}\le l(x)\Big|\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2\right)\nonumber\\[3pt] &\le\textrm{P}\!\left(Z_{11}{\,\gt\,}x-l(x)\Big|\theta_{11}=s_{11},D_{11}=u_{11}\right)\nonumber\\[3pt] &=\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}{\,\gt\,}x-l(x)\Big|\theta_{11}=s_{11}\right)\mathbb I_{\{s_{11}+u_{11}{\,\gt\,}t\}}\nonumber\\[3pt] &\quad +\textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(s_{11})}+ Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,} x - l(x)\Big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\mathbb I_{\!\left\{s_{11}+ u_{11}\le t\right\}}\nonumber\\[3pt] &\quad \sim \textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x -l(x)\right) h_1(s_{11})\nonumber \\[3pt] &\quad +\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})} {\,\gt\,} x -l(x)\right)\varphi_1(s_{11},u_{11})\mathbb I_{\!\left\{s_{11}+u_{11}\le t\right\}}\nonumber\\[3pt] &\quad \sim \textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(s_{11})}{\,\gt\,} x\right) h_1(s_{11})+\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,} x \right)\varphi_1(s_{11},u_{11})\mathbb I_{\!\left\{s_{11}+ u_{11}\le t\right\}}\nonumber\\[3pt] &\quad \sim \textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x\Big|\theta_{11}=s_{11}\right)\nonumber\\[3pt] &\quad +\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left\{s_{11}+u_{11}\le t\right\}} {\,\gt\,} x\Big|\theta_{11}=s_{11},D_{11}=u_{11}\right),\end{align}
where we use Assumptions 2.2 and 2.1 in the fourth step and the last step, and in the fifth step we employ Lemma 4.1.
Similarly, we have
\begin{align}I_3(x,t)&=\textrm{P}(Z_{11}+Z_{21}{\,\gt\,}x,Z_{21}{\,\gt\,} x-l(x)\mid\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2)\nonumber\\[3pt] &\le\textrm{P}(Z_{21}{\,\gt\,}x-l(x)\mid\theta_{21}=s_{21},D_{21}=u_{21})\nonumber\\[3pt] &\quad \sim \textrm{P} \!\left( X_{21}\textrm{e}^{-R_2(s_{21})}{\,\gt\,}x\big|\theta_{21}=s_{21}\right)\nonumber\\[3pt] &\quad +\textrm{P} \!\left( Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{s_{21}+u_{21}\le t\right\}}{\,\gt\,}x\big|\theta_{21}=s_{21},D_{21}=u_{21}\right)\end{align}
holds uniformly for
$s_{k1},u_{k1},t\in \Lambda_T,k=1,2$
.
As for
$I_2(x,t)$
, applying Assumption 2.2 in the third step, we have, for sufficiently large x and
$\varepsilon=\varepsilon(T)$
,
\begin{align*}I_2(x,t)&=\textrm{P}(Z_{11}+Z_{21} {\,\gt\,} x,l(x) {\,\lt\,} Z_{21}\le x-l(x)\mid\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2)\nonumber\\[5pt] &=\int_{l(x)}^{x-l(x)}\textrm{P}\!\left( Z_{11} {\,\gt\,} x - y\mid\theta_{11}=s_{11},D_{11}=u_{11}\right)\textrm{P}( Z_{21}\in \mathrm d y |\theta_{21}=s_{21},D_{21}=u_{21})\nonumber\\[5pt] &\le (1+\varepsilon)\int_{l(x)}^{x-l(x)} \!\left(h_1(s_{11})\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x- y \right)\right.\nonumber\\[5pt] &\quad \!\left.+\varphi_1(s_{11},u_{11})\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left\{ s_{11}+u_{11}\le t \right\}} {\,\gt\,} x-y \right)\right)\nonumber\\[5pt] &\quad \!\left( h_2(s_{21})\textrm{P} \!\left( X_{21}\textrm{e}^{-R_2(s_{21})}\in \mathrm d y \right)+\varphi_2\!\left( s_{21},u_{21}\right)\textrm{P} \!\left( Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{ s_{21}+u_{21}\le t \right\}}\in \mathrm d y \right)\right)\nonumber\\[5pt] &=(1+\varepsilon)h_1(s_{11})h_2(s_{21})\nonumber\\[5pt] &\quad\, \textrm{P}\! \left(X_{11}\textrm{e}^{-R_1(s_{11})}+X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x,l(x) {\,\lt\,} X_{21}\textrm{e}^{-R_2(s_{21})}\le x-l(x)\right)\nonumber\\[5pt] &\quad +(1+\varepsilon)\varphi_1(s_{11},u_{11}) h_2(s_{21})\\ &\quad \times \textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left\{s_{11}+u_{11}\le t\right\}} + X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x,l(x) {\,\lt\,} X_{21}\textrm{e}^{-R_2(s_{21})}\le x- l(x)\right)\nonumber\\[5pt] &\quad +(1+\varepsilon) h_1(s_{11})\varphi_2(s_{21},u_{21})\textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(s_{11})}+ Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{s_{21}+ u_{21}\le t\right\}} {\,\gt\,} x,\right.\nonumber\\[5pt] &\times\!\left.l(x) {\,\lt\,} Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{ s_{21}+u_{21}\le t \right\}}\le x-l(x)\right)\nonumber\\[5pt] &\quad +(1+\varepsilon)\varphi_1(s_{11},u_{11})\varphi_2(s_{21},u_{21})\nonumber\\[5pt] &\quad\, \textrm{P} \!\left(Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left\{s_{11}+u_{11}\le t\right\}}+Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{s_{21}+u_{21}\le t\right\}} {\,\gt\,} x,\right.\nonumber\\[5pt] &\quad \times \!\left.l(x) {\,\lt\,} Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{s_{21}+u_{21}\le t\right\}}\le x-l(x)\right)\nonumber\\[5pt] &\;:\!=\;(1+\varepsilon)(I_{21}+I_{22}+I_{23}+I_{24})\end{align*}
holds uniformly for
$s_{11},u_{11},s_{21},u_{21},t\in \Lambda_T$
. We deal with
$I_{21}$
first. For sufficiently large x and
$\varepsilon_*=\varepsilon_*(T)$
, using Theorem 1 of [Reference Tang and Yuan24] in the second step, we obtain
\begin{align}&\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}+X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x,l(x) {\,\lt\,} X_{21}\textrm{e}^{-R_2(s_{21})}\le x-l(x)\right)\nonumber\\[5pt] & \le \textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}+X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x\right)-\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x,X_{21}\textrm{e}^{-R_2(s_{21})}\le l(x)\right)\nonumber\\[5pt] & \quad-\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}\le l(x),X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x\right)\nonumber\\[5pt] & \le (1+\varepsilon_*)\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x\right)+\textrm{P}\!\left(X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x\right)\right)\nonumber\\[5pt] &\quad-\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x\right)\textrm{P}\!\left(X_{21}\textrm{e}^{-R_2(s_{21})}\le l(x)\right)\nonumber\\[5pt] & \quad-\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}\le l(x)\right)\textrm{P}\!\left(X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x\right)\nonumber\\[5pt] & \le 2\varepsilon_*P\!\left(X_{11}\textrm{e}^{-R_1(s_{11})} {\,\gt\,} x\right)+2\varepsilon_*P\!\left(X_{21}\textrm{e}^{-R_2(s_{21})} {\,\gt\,} x\right)\end{align}
holds uniformly for
$s_{11},s_{21}\in \Lambda_T$
. This, together with (6) shows that
\begin{align}I_{21}&= h_1(s_{11})h_2(s_{21})\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}+ X_{21}\textrm{e}^{-R_2(s_{21})}{\,\gt\,} x,l(x){\,\lt\,} X_{21}\textrm{e}^{-R_2(s_{21})}\le x - l(x)\right)\nonumber\\[2pt] &=o\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}{\,\gt\,}x\big| \theta_{11}=s_{11}\right)+\textrm{P}\!\left(X_{21}\textrm{e}^{-R_2(s_{21})}{\,\gt\,}x\big|\theta_{21}=s_{21}\right)\right).\end{align}
By the same idea used previously, it can be deduced that
\begin{align}I_{22}=o&\!\left(\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left\{s_{11}+u_{11}\le t\right\}}{\,\gt\,}x\big| \theta_{11}=s_{11},D_{11}=u_{11}\right)\right.\nonumber\\[2pt] &\;\!\left.+\textrm{P}\!\left(X_{21}\textrm{e}^{-R_2(s_{21})}{\,\gt\,}x\big|\theta_{21}=s_{21}\right)\right),\end{align}
\begin{align}I_{23}=o &\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}{\,\gt\,}x\big| \theta_{11}=s_{11}\right)\right.\nonumber\\[2pt] &\!\left.+\textrm{P}\!\left(Y_{21}\textrm{e}^{-R_2(s_{21}+u_{21})}\mathbb I_{\!\left\{s_{21}+u_{21}\le t\right\}}{\,\gt\,}x\big|\theta_{21}=s_{21},D_{21}=u_{21}\right)\right)\end{align}
and
\begin{align}I_{24}=o\!\left(\sum\limits_{k=1}^2\textrm{P}\!\left(Y_{k1}\textrm{e}^{-R_k(s_{k1}+u_{k1})}\mathbb I_{\!\left\{s_{k1}+u_{k1}\le t\right\}}{\,\gt\,}x\big| \theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)\right)\end{align}
hold uniformly for
$s_{11},u_{11},s_{21},u_{21},t\in \Lambda_T$
.
Substituting (26)–(29) back into (25) leads that
\begin{align}I_{2}(x,t)=o&\Bigg(\sum\limits_{k=1}^2 \bigg[\textrm{P} \!\left(X_{k1}\textrm{e}^{-R_k(s_{k1})}{\,\gt\,}x\big|\theta_{k1}=s_{k1}\right)\nonumber\\[2pt] &+\textrm{P} \!\left(Y_{k1}\textrm{e}^{-R_k(s_{k1}+u_{k1})}\mathbb I_{\!\left\{s_{k1}+u_{k1}\le t\right\}}{\,\gt\,}x\big|\theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)\bigg]\Bigg)\end{align}
holds uniformly for
$s_{11},u_{11},s_{21},u_{21},t\in \Lambda_T$
.
A combination of (23), (24), and (30) derives the upper-bound version of (22).
The corresponding lower-bound version of (22) can be easily established. In fact, by applying Assumption 2.2, followed by 2.1 in the last step, we have
\begin{align}&\textrm{P}\!\left(Z_{11}+Z_{21}{\,\gt\,}x\big|\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2\right)\nonumber\\[2pt] & \ge \sum\limits_{k=1}^2\textrm{P}\!\left(Z_{k1}{\,\gt\,}x\big|\theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)-\prod\limits_{k=1}^2 \textrm{P}\!\left(Z_{k1}{\,\gt\,}x\big|\theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)\nonumber\\[2pt] & {\,\gtrsim\,} \sum\limits_{k=1}^2 \bigg[\textrm{P}\!\left(X_{k1}\textrm{e}^{-R_k(s_{k1})}{\,\gt\,} x\big|\theta_{k1}=s_{k1}\right)\nonumber\\[2pt] & \quad+\textrm{P}\!\left(Y_{k1}\textrm{e}^{-R_k(s_{k1}+u_{k1})}\mathbb I_{\!\left\{s_{k1}+u_{k1}\le t\right\}}{\,\gt\,} x\big| \theta_{k1}=s_{k1},D_{k1}=u_{k1}\right)\bigg]\end{align}
holds uniformly for
$s_{11},u_{11},s_{21},u_{21},t\in \Lambda_T$
. Combining these estimates completes the proof of (22). Next, we proceed by virtue of induction that for every
$n_2\in\mathbb N_+$
,
\begin{align}&\textrm{P}\!\left(Z_{11}+\sum\limits_{i=1}^{n_2}Z_{2i}{\,\gt\,}x\big|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[5pt] &\sim\textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)+\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(t_{11}+u_{11})}\mathbb I_{\!\left(t_{11}+u_{11}\le t\right)}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\nonumber\\[5pt] &\quad +\sum\limits_{i=1}^{n_2}\bigg[\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\nonumber\\[5pt] & \quad+\textrm{P}\!\left(Y_{2i}\textrm{e}^{-R_2(t_{2i}+u_{2i})}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t\right\}}{\,\gt\,}x\big|\theta_{2i}=s_{2i},D_{2i}=u_{2i}\right)\bigg]\end{align}
holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i},t\in\Lambda_T$
,
$1\le i\le n$
. Obviously, (22) indicates that the assertion holds for
$n_2=1$
. Now we assume by induction that (32) holds for
$n_2-1$
,
\begin{align}&\textrm{P}\!\left(Z_{11}+\sum\limits_{i=1}^{n_2-1}Z_{2i}{\,\gt\,}x\big|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2-1\right)\nonumber\\[5pt] & \sim \textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)+\textrm{P} \!\left( Y_{11}\textrm{e}^{-R_1(t_{11}+u_{11})}\mathbb I_{\!\left(t_{11}+u_{11}\le t\right)}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\nonumber\\[5pt] & \quad+\sum\limits_{i=1}^{n_2-1}\bigg[\textrm{P} \!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\nonumber\\[5pt] & \quad +\textrm{P} \!\left(Y_{2i}\textrm{e}^{-R_2(t_{2i}+u_{2i})}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t\right\}}{\,\gt\,}x\big|\theta_{2i}=s_{2i},D_{2i}=u_{2i}\right)\bigg].\end{align}
It remains to show that relation (32) holds for
$n_2$
. Depending on the value of
$Z_{2n_2}$
belonging to
$\!\left(0,l(x)\right]$
,
$\!\left(l(x),x-l(x)\right]$
, and
$(x-l(x),\infty)$
, we split the probability on the left-hand side of (32) into three parts as follows:
\begin{align}&\textrm{P}\!\left(Z_{11}+\sum\limits_{i=1}^{n_2}Z_{2i}{\,\gt\,}x\Bigg|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[5pt] &\quad\quad = I_4(x,n_2,t)+I_5(x,n_2,t)+I_6(x,n_2,t).\end{align}
For
$I_4(x,n_2,t)$
, as dealing with
$I_1(x,t)$
, by the induction assumption together with Lemma 4.1, it holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
, that
\begin{align} &I_4(x,n_2,t)\nonumber \\&=\textrm{P}\!\left( Z_{11}+\sum\limits_{i=1}^{n_2} Z_{2i}{\,\gt\,} x,Z_{2n_2}\le l(x)\big|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i \le n_2\right)\nonumber\\[5pt] &\le\textrm{P}\!\left( Z_{11}+\sum\limits_{i=1}^{n_2-1} Z_{2i}{\,\gt\,} x - l(x)\big| \theta_{11}=s_{11},D_{11}=u_{11}, \theta_{2i}=s_{2i}, D_{2i}=u_{2i},1\le i\le n_2-1 \right)\nonumber\\[5pt] &\sim \textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)+\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(t_{11}+u_{11})}\mathbb I_{\!\left(t_{11}+u_{11}\le t\right)}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\nonumber\\[5pt] &+\sum\limits_{i=1}^{n_2-1}\bigg[\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}= s_{2i}\right)\nonumber\\[5pt] &+\textrm{P}\!\left(Y_{2i}\textrm{e}^{-R_2(t_{2i}+u_{2i})}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t\right\}}{\,\gt\,}x\big|\theta_{2i}= s_{2i},D_{2i}= u_{2i}\right)\bigg].\end{align}
At the same time, employing Assumption 2.2 and Lemma 4.1 we obtain
\begin{align}& I_6(x,n_2,t)\nonumber\\&=\textrm{P}\!\left(\! Z_{11}+\sum\limits_{i=1}^{n_2}Z_{2i}{\,\gt\,} x,Z_{2n_2}{\,\gt\,} x - l(x)\big|\theta_{11}= s_{11},D_{11}= u_{11},\theta_{2i}= s_{2i},D_{2i}= u_{2i},1\le i \le n_2\!\right)\nonumber\\[5pt] &\le\textrm{P}\!\left( Z_{2n_2}{\,\gt\,}x-l(x)\big| \theta_{2n_2}=s_{2n_2},D_{2n_2}=u_{2n_2}\right)\nonumber\\[5pt] & \sim \textrm{P} \!\left( X_{2n_2}\textrm{e}^{-R_2\!\left(t_{2n_2}\right)}{\,\gt\,} x\big| \theta_{2n_2}= s_{2n_2}\right)\nonumber\\[5pt] &\quad +\textrm{P}\!\left(Y_{2n_2}\textrm{e}^{-R_2\!\left( t_{2n_2}+u_{2n_2}\right)}\mathbb I_{\!\left\{t_{2n_2}+u_{2n_2}\le t \right\}}{\,\gt\,}x\big| \theta_{2n_2}=s_{2n_2},D_{2n_2}= u_{2n_2}\right)\end{align}
holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
.
In the following, we focus on
$I_5(x,n_2,t)$
. Given that
$R_k(t)$
is nonnegative, we use the notation
$\textrm{e}^{-\hat R}\;:\!=\;\max\{\textrm{e}^{-R_1(t_{11})},\textrm{e}^{-R_2(t_{21})}\}$
. It holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i},t\in\Lambda_T$
,
$1\le i\le n_2$
, that
\begin{align} &\!\!\!\!\!\!\! I_5(x,n_2,t) \nonumber \\&=\textrm{P}\Bigg( Z_{11}+\sum\limits_{i=1}^{n_2}Z_{2i}{\,\gt\,} x,l(x) {\,\lt\,} Z_{2n_2}\le x - l(x)\big|\theta_{11}=s_{11},D_{11}=u_{11},\nonumber\\[5pt] &\qquad\quad \theta_{2i}=s_{2i},D_{2i}= u_{2i},1\le i \le n_2\Bigg)\nonumber\\[5pt] &\le\textrm{P} \Bigg(Z_{11}+\sum\limits_{i=1}^{n_2} Z_{2i}{\,\gt\,}x,Z_{11}+\sum\limits_{i=1}^{n_2-1} Z_{2i}{\,\gt\,}l(x),Z_{2n_2}{\,\gt\,}l(x)\big|\theta_{11}=s_{11},D_{11}=u_{11},\nonumber\\[5pt] &\qquad\quad \theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2 \Bigg)\nonumber\\[5pt] &\le\int_0^1\textrm{P} \bigg(X_{11}+Y_{11}\mathbb I_{\{t_{11}+u_{11}\le t\}}+\sum\limits_{i=1}^{n_2}X_{2i}+Y_{2i}\mathbb I_{\!\left\{ t_{2i}+u_{2i}\le t\right\}}{\,\gt\,}\frac{x}{v},\nonumber\\[5pt] &\qquad\quad X_{11}+Y_{11}\mathbb I_{\{t_{11}+u_{11}\le t\}}+\sum\limits_{i=1}^{n_2-1}X_{2i}+Y_{2i}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t \right\}}{\,\gt\,}\frac{l(x)}{v},\nonumber\\[5pt] &\qquad\quad X_{2n_2}+ Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}{\,\gt\,} \frac{l(x)}{v}\big|\theta_{11}= s_{11},D_{11}= u_{11},\!\theta_{2i}= s_{2i},D_{2i}= u_{2i},\!1\le i \le n_2\! \!\bigg)\nonumber\\[5pt] &\textrm{P} \!\left( \textrm{e}^{-\hat R}\in\textrm{d} v \right)\nonumber\\[5pt] &=\int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P} \!\left(X_{11}+Y_{11}\mathbb I_{\{t_{11}+u_{11}\le t\}}+\sum\limits_{i=1}^{n_2-1}X_{2i}+Y_{2i}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t \right\}}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\big|\right.\nonumber\\[5pt] &\theta_{11}= s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2-1\bigg)\nonumber\\[5pt] &\textrm{P} \!\left(X_{2n_2}+Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}} \in \textrm{d} z\big|\theta_{2n_2}= s_{2n_2},D_{2n_2}=u_{2n_2}\right)\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &\sim \int_0^1\int_{\frac{l(x)}{v}}^{\infty}\Bigg(\textrm{P} \!\left(X_{11}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\big| \theta_{11}=s_{11}\right)\nonumber\\[5pt] &+\textrm{P} \!\left(Y_{11}\mathbb I_{\!\left\{t_{11}+u_{11}\le t \right\}}{\,\gt\,}\!\left(\frac{x}{v}- z\right)\vee\frac{l(x)}{v}\big| \theta_{11}=s_{11},D_{11}=u_{11}\right)\nonumber\\[5pt] &+\sum\limits_{i=1}^{n_2-1}\bigg[\textrm{P}\!\left(X_{2i}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\big| \theta_{2i}=s_{2i}\right)\nonumber\\[5pt] &+\textrm{P}\!\left(Y_{2i}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t \right\}}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\big| \theta_{2i}=s_{2i},D_{2i}=u_{2i}\right)\bigg]\Bigg)\nonumber\\[5pt] &\!\left(\textrm{P}\!\left(X_{2n_2}\in \textrm{d} z\right)h_2(s_{2n_2})+\textrm{P}\!\left(Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}\in \textrm{d} z\right)\varphi_2(s_{2n_2},u_{2n_2})\right)\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] \;:\!=\;&\sum\limits_{k=1}^8 I_{5k},\end{align}
where we use (33) and Assumption 2.2 in the penultimate step. From Theorem 1 of [Reference Tang and Yuan24] we get that for sufficiently large x, there exists some constant
$\varepsilon{\,\gt\,}0$
such that
\begin{align}&\textrm{P}\!\left(X_{11}+X_{2n_2}{\,\gt\,}x,X_{11}{\,\gt\,}l(x),X_{2n_2}{\,\gt\,}l(x)\right)\nonumber\\[5pt] &\quad\le \textrm{P}\!\left(X_{11}+X_{2n_2}{\,\gt\,}x\right)-\textrm{P}\!\left(X_{11}{\,\gt\,}x,X_{2n_2}\le l(x)\right)-\textrm{P}\!\left(X_{11}\le l(x),X_{2n_2}{\,\gt\,}x\right)\nonumber\\[5pt] &\quad \le \varepsilon\!\left(\textrm{P}\!\left(X_{11}{\,\gt\,}x\right)+\textrm{P}\!\left(X_{2n_2}{\,\gt\,}x\right)\right).\end{align}
Consequently, employing (38) in the fourth step we obtain
\begin{align}I_{51}&=\int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P}\!\left(X_{11}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\big| \theta_{11}= s_{11}\right)\textrm{P}\!\left(X_{2n_2}\in \textrm{d} z\right)h_2(s_{2n_2})\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &\sim \int_0^1 h_1(s_{11}) h_2(s_{2n_2})\textrm{P}\!\left( X_{11}+ X_{2n_2}{\,\gt\,} \frac{x}{v},X_{11}{\,\gt\,}\frac{l(x)}{v},X_{2n_2}{\,\gt\,}\frac{l(x)}{v}\right)\textrm{P}\!\left( \textrm{e}^{-\hat R}\in\textrm{d} v \right)\nonumber\\[5pt] &\le\int_0^1 h_1(s_{11}) h_2(s_{2n_2})\textrm{P}\!\left( X_{11}+ X_{2n_2}{\,\gt\,} \frac{x}{v},X_{11}{\,\gt\,} l\!\left(\frac{x}{v}\right),X_{2n_2}{\,\gt\,} l\!\left(\frac{x}{v}\right)\right)\textrm{P}\!\left( \textrm{e}^{-\hat R}\in\textrm{d} v \right)\nonumber\\[5pt] &\le\varepsilon h_1(s_{11})h_2(s_{2n_2})\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-\hat R}{\,\gt\,}x\right)+\textrm{P}\!\left(X_{2n_2}\textrm{e}^{-\hat R}{\,\gt\,}x\right)\right)\nonumber\\[5pt] &\le\varepsilon h_1(s_{11})h_2(s_{2n_2})\bigg(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\right)+\textrm{P}\!\left(X_{11}\textrm{e}^{-R_2(t_{21})}{\,\gt\,} x\right)\nonumber\\[5pt] &+\textrm{P}\!\left(X_{2n_2}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\right)+\textrm{P}\!\left(X_{2n_2}\textrm{e}^{-R_2(t_{21})}{\,\gt\,} x\right)\bigg)\nonumber\\[5pt] &=o\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)+\sum\limits_{i=1}^{n_2}\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\right)\end{align}
holds uniformly for
$s_{11},s_{2n_2}\in\Lambda_T$
, where in the last step we use Assumption 2.1 and the fact that if
$\overline F_1(x)\asymp\overline F_2(x)$
then
$\overline F_1\!\left(\textrm{e}^{R_1(t)}x\right)\asymp\overline F_2\!\left(\textrm{e}^{R_1(t)}x\right)$
for
$0 {\,\lt\,} \textrm{e}^{-R_k(t)}\le 1$
a.s. Analogously, given that
$\overline F_1\asymp\overline G_1$
and
$\overline F_2\asymp\overline G_2$
, there exist some constants
$M_1,M_2{\,\gt\,}0$
, irrespective of
$s_{11},u_{11},s_{2n_2},u_{2n_2}$
such that
\begin{align}I_{54}&=\int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P}\!\left(Y_{11}\mathbb I_{\!\left\{t_{11}+u_{11}\le t \right\}}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\bigg| \theta_{11}=s_{11},D_{11}=u_{11}\right)\nonumber\\[5pt] &\quad \textrm{P}\!\left(Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}\in \textrm{d} z\right)\varphi_2(s_{2n_2},u_{2n_2})\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &\sim \varphi_1(s_{11},u_{11})\varphi_2(s_{21},u_{21})\nonumber\\[5pt] &\quad \int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P} \!\left( Y_{11}\mathbb I_{\!\left\{t_{11}+u_{11}\le t \right\}}{\,\gt\,} \!\left(\frac{x}{v}- z \right)\vee\frac{l(x)}{v}\right)\textrm{P} \!\left( Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}\in \textrm{d} z \right)\textrm{P} \!\left( \textrm{e}^{-\hat R}\in\textrm{d} v \right)\nonumber\\[5pt] &\le M_1\varphi_1(s_{11},u_{11})\varphi_2(s_{21},u_{21})\nonumber\\[5pt] &\quad \int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P}\!\left(X_{11}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\right)\textrm{P}\!\left(Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}\in \textrm{d} z\right)\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &=M_1\varphi_1(s_{11},u_{11})\varphi_2(s_{21},u_{21})\nonumber\\[5pt] &\quad \int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P}\!\left(Y_{2n_2}\mathbb I_{\!\left\{ t_{2n_2}+u_{2n_2}\le t \right\}}{\,\gt\,}\!\left(\frac{x}{v}-z\right)\vee\frac{l(x)}{v}\right)\textrm{P}\!\left(X_{11}\in \textrm{d} z\right)\textrm{P}\!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &\le M_1M_2\varphi_1(s_{11},u_{11})\varphi_2(s_{21},u_{21})\nonumber\\[5pt] &\quad \int_0^1\int_{\frac{l(x)}{v}}^{\infty}\textrm{P}\!\left(X_{2n_2}{\,\gt\,}\!\left(\frac{x}{v}- z\right)\vee\frac{l(x)}{v}\right)\textrm{P}\!\left(X_{11}\in\textrm{d} z\right)\textrm{P} \!\left(\textrm{e}^{-\hat R}\in\textrm{d} v\right)\nonumber\\[5pt] &=o\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)+\sum\limits_{i=1}^{n_2}\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\right)\end{align}
holds uniformly for
$s_{11},u_{11},s_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
. Subsequently, we have for
$1\le k\le8$
,
\begin{align}I_{5k}=o\!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)+\sum\limits_{i=1}^{n_2}\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\right)\end{align}
hold uniformly for
$s_{11},u_{11},s_{2i},u_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
. Due to the repetitiveness of the technical method used, we omit the detailed proof.
Combining (37) and (41) yields that
\begin{align} I_5(x,n_2,t)&=o\Bigg(\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(t_{11})}{\,\gt\,} x\big|\theta_{11}=s_{11}\right)+\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(t_{11}+u_{11})}\mathbb I_{\!\left(t_{11}+u_{11}\le t\right)}{\,\gt\,} x\big|\theta_{11}=s_{11}\right)\nonumber\\[5pt] &+\sum\limits_{i=1}^{n_2}\bigg[\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,} x\big|\theta_{2n_2} = s_{2n_2}\right)\nonumber\\[5pt] &+\textrm{P} \!\left(Y_{2i}\textrm{e}^{-R_2(s_{2i}+u_{2i})}\mathbb I_{\!\left(t_{2i}+u_{2i}\le t\right)} {\,\gt\,} x\big|\theta_{2i} = s_{2i},D_{2i} = u_{2i}\right)\bigg]\Bigg)\end{align}
holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
. This, along with (34)–(36) gives the upper bound of (32).
Now we turn to the lower bound of (32), that is
\begin{align}&\textrm{P}\!\left(Z_{11}+\sum\limits_{i=1}^{n_2}Z_{2i}{\,\gt\,}x\big|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[5pt] &\quad\ge \textrm{P}\!\left(Z_{11}+\sum\limits_{i=1}^{n_2-1}Z_{2i}{\,\gt\,}x\big| \theta_{11}=s_{11},D_{11}=u_{11}, \theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2-1\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(Z_{2n_2}{\,\gt\,}x\big| \theta_{2n_2}=s_{2n_2},D_{2n_2}=u_{2n_2}\right)\nonumber\\[5pt] &\quad\quad-\textrm{P} \!\left( Z_{11}+\sum\limits_{i=1}^{n_2-1} Z_{2i}{\,\gt\,} x,Z_{2n_2}{\,\gt\,} x\big| \theta_{11}=s_{11},D_{11}=u_{11}, \theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i \le n_2\right)\nonumber\\[5pt] &\quad\sim \textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(s_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)+\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(s_{11}+u_{11})}\mathbb I_{\!\left(s_{2i}+u_{2i}\le T\right)}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)\nonumber\\[5pt] &\quad\quad+\sum\limits_{i=1}^{n_2}\bigg(\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(t_{2i})}{\,\gt\,}x\big|\theta_{2i}=s_{2i}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(Y_{2i}\textrm{e}^{-R_2(t_{2i}+u_{2i})}\mathbb I_{\!\left\{t_{2i}+u_{2i}\le t\right\}}{\,\gt\,}x\big|\theta_{2i}=s_{2i},D_{2i}=u_{2i}\right)\bigg)\end{align}
holds uniformly for
$s_{11},u_{11},s_{2i},u_{2i}\in\Lambda_T$
,
$1\le i\le n_2$
. Thus (32) is verified. Finally, for any fixed
$n_2$
, we turn to deduce (21) by applying induction for
$n_1$
again. Proceeding in a similar way as when deriving (32), coupled with the independence of
$(X_{ki},Y_{ki},\theta_{ki},D_{ki})$
,
$1\le i\le n_k$
,
$k=1,2$
, we complete the proof.
Subsequently, for a distribution
$F\in\mathcal S$
, the well-known Kesten inequality is a useful means in dealing with the tail probability of randomly weighted sums of infinitely terms. Readers can see Lemma 4.4 of [Reference Li, Tang and Wu20], Lemma A.3 of [Reference Yuan and Lu32], and Theorem 1.2 of [Reference Chen4] for more different forms of the inequality regarding the various dependence structures of the risk models. In this paper, comparing with the commonly used version of Kesten inequality, we establish an inequality of conditional Kesten type.
Lemma 4.3. Consider the bidimensional risk model (1) with
$F_k,G_k\in\mathcal S$
,
$k=1,2$
, and
$\overline F_i(x)\asymp\overline G_j(x)$
,
$i,j=1,2$
. Under the conditions of Theorem 2.1, we have for any
$\varepsilon{\,\gt\,}0$
,
$T\in\Lambda$
, there exists some constant
$K=K(\varepsilon,T){\,\gt\,}0$
such that
\begin{align}&\textrm{P}\bigg(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\theta_{k1})}+ Y_{ki}\textrm{e}^{-R_k(\theta_{k1}+D_{k1})}\mathbb I_{\!\left\{\theta_{ki}+ D_{ki}\le t\right\} }{\,\gt\,} x\Bigg|\theta_{ki}\nonumber\\&\quad = s_{ki},D_{ki}= u_{ki},1\le i \le n_k,k=1,2\Bigg)\nonumber\\[5pt] &\quad\le K{(1+\varepsilon)}^{n_1+n_2}\bigg(\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,}x\big|\theta_{11}=s_{11},\theta_{21}=s_{21}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+ Y_{21}\textrm{e}^{-R_2(\theta_{21}+D_{21})}\mathbb I_{\!\left\{\theta_{21}+D_{21}\le t\right\}}{\,\gt\,} x\big|\theta_{11}=s_{11},\theta_{21}=s_{21},D_{21}=u_{21}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P} \!\left( Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}+ X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,} x\big|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{21}=s_{21}\right)\nonumber\\[5pt] &\quad\quad\!\left.+\textrm{P}\!\left(\sum\limits_{k=1}^2Y_{k1}\textrm{e}^{-R_k(\theta_{k1}+D_{k1})}\mathbb I_{\!\left\{\theta_{k1}+D_{k1}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2\right)\right)\end{align}
holds uniformly for
$n_k=1,2,\ldots$
,
$k=1,2$
,
$x{\,\gt\,}0$
, and
$s_{ki},u_{ki},t\in\Lambda_T$
,
$1\le i\le n_k$
,
$k=1,2$
.
Proof. We first show a unidimensional form of (44) with constant weights, that is, for every given
$\varepsilon{\,\gt\,}0$
, there exists a constant
$M=M(\varepsilon,a,b,T)$
such that
\begin{align}&\textrm{P}\!\left(\sum\limits_{i=1}^{n} vX_i+wY_i\mathbb I_{\!\left\{\theta_i+D_i\le t\right\}}{\,\gt\,}x\Bigg|\theta_i=s_i,D_i=u_i,1\le i\le n\right)\nonumber\\[5pt] &\quad \le M{(1+\varepsilon)}^{n}\!\left(\textrm{P}\!\left(vX_1{\,\gt\,}x\big|\theta_1=s_1\right)+\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}{\,\gt\,}x\big|\theta_1=s_1,D_1=u_1\right)\right)\end{align}
holds uniformly for
$n=1,2,\ldots$
,
$x{\,\gt\,}0$
,
$s_{i},u_{i},t\in\Lambda_T$
,
$1\le i\le n$
, and constants
$v,w\in[a,b]$
with
$0 {\,\lt\,} a\le b{\,\lt\,}\infty$
. By (6) in Assumption 2.1, there exists some constant
$x_0{\,\gt\,}0$
such that for all
$t\in\Lambda_T$
and
$v\in [a,b]$
,
\begin{align}\textrm{P}\!\left(vX_{1}{\,\gt\,}x_0\big|\theta_{1}=t\right)\ge\frac{1}{2}{\overline F}_1\!\left(x_0/a\right)h(t).\end{align}
Meanwhile, according to Assumption 2.2, for every
$\varepsilon_1{\,\gt\,}0$
and all
$x{\,\gt\,}x_0$
, we have
\begin{align}&\textrm{P}\!\left(vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\nonumber\\[5pt] &\quad\ge (1+\varepsilon_1)\big[ \textrm{P}\!\left(vX_{n+1}{\,\gt\,} x\big|\theta_{n+1}=s_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,} x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\big].\end{align}
We write
\begin{align}\alpha_n= \sup_{\substack{x \ge 0,t\in \Lambda_T\\ v,w\in[a,b]}}\mathop{\sup}_{\substack{s_{i},u_{i}\in\Lambda_T\\ 1\le i\le n}}\frac {\textrm{P}\!\left(\sum\limits_{i=1}^{n} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}} {\,\gt\,} x\bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i \le n\right)}{\textrm{P}\!\left(vX_1 {\,\gt\,} x\big|\theta_1=s_1\right)+\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}} {\,\gt\,} x\big|\theta_1=s_1,D_1=u_1\right)}.\end{align}
Subsequently, we start evaluating
$\alpha_{n+1}$
. It is clear that
\begin{align}&\textrm{P}\!\left(\sum\limits_{i=1}^{n+1} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n\right)\nonumber\\[5pt] &\quad= \textrm{P}\bigg(\sum\limits_{i=1}^{n+1} vX_{i}+ wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}{\,\gt\,} x,vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\nonumber\\ &\quad\quad \le x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i \le n+1\Bigg)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{1n+1}=u_{n+1}\right).\end{align}
For every
$x{\,\gt\,}x_0$
, conditioning on
$vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}$
yields that
\begin{align}&\textrm{P}\Bigg(\sum\limits_{i=1}^{n+1} vX_{i}+ wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}} {\,\gt\,} x,vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\nonumber\\[5pt]&\le x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i}, 1\le i\le n+1\Bigg)\nonumber\\[5pt] &\quad= \int_0^x\textrm{P}\!\left(\sum\limits_{i=1}^{n} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}} {\,\gt\,} x-y\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i \le n\right)\nonumber\\[5pt] &\textrm{P}\!\left(vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\in\textrm{d} y\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\nonumber\\[5pt] &\quad= \int_0^x \frac{\textrm{P}\!\left(\sum\limits_{i=1}^{n} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}} {\,\gt\,} x-y\bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n\right)}{\textrm{P}\!\left(vX_1 {\,\gt\,} x-y\big|\theta_1=s_1\right)+\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}} {\,\gt\,} x-y\big|\theta_1=s_1,D_1=u_1\right)}\nonumber\\[5pt] &\quad\quad\times\big[\textrm{P}\!\left(vX_1 {\,\gt\,} x-y\big|\theta_1=s_1\right)+\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}} {\,\gt\,} x-y\big|\theta_1=s_1,D_1=u_1\right)\big]\nonumber\\[5pt] &\quad\quad\times\textrm{P}\!\left(vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\in\textrm{d} y\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\nonumber\\[5pt] &\le \alpha_n\Big[\textrm{P}\!\left(vX_1+vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}} {\,\gt\,} x,\right. \nonumber\\[5pt] &\quad\quad\!\left. vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\le x\big|\theta_{1}=s_1,\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}+vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}} {\,\gt\,} x,\right.\nonumber\\[5pt] &\quad\quad\!\left.vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\le x\big|\theta_{1}= s_1,D_1= u_1,\theta_{n+1}= s_{n+1},D_{n+1}= u_{n+1}\right)\Big]\nonumber\\[5pt] &\quad=\alpha_n\Big[\textrm{P}\!\left( vX_1+ vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}} {\,\gt\,} x\big|\theta_{1}= s_1,\theta_{n+1}= s_{n+1},D_{n+1}= u_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\Big(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}+ vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\nonumber \\ &\quad\quad {\,\gt\,} x\big|\theta_{1}=s_1,D_1=u_1,\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\Big)\nonumber\\[5pt] &\quad\quad-2\textrm{P}\!\left(vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}} {\,\gt\,} x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\Big].\end{align}
Proceeding in the same way as Lemma A.2 of [Reference Yuan and Lu32], it can be shown that
\begin{align}&\textrm{P}\!\left(vX_1+vX_{n+1}+wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}} {\,\gt\,} x\big|\theta_{1}=s_1,\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\nonumber\\[5pt] &\quad\sim \textrm{P}\!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}=s_1\right)+\textrm{P}\!\left(vX_{n+1}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\end{align}
and
\begin{align}&\textrm{P}\Big(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}} + vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\nonumber \\&\quad {\,\gt\,} x\big|\theta_{1} = s_1,D_1 = u_1,\theta_{n+1} =s_{n+1},D_{n+1} =u_{n+1}\Big)\nonumber\\[5pt] &\quad\sim \textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}{\,\gt\,}x\big|\theta_{1}=s_1,D_1=u_1\right)+\textrm{P}\!\left(vX_{n+1}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P}\!\left(wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\end{align}
hold uniformly for
$s_1,u_1,s_{n+1},u_{n+1}\in \Lambda_T$
and
$v,w\in[a,b]$
. According to
$X_1$
belong to
$\!\left(0,l(x)\right]$
,
$\!\left(l(x),x-l(x)\right]$
, and
$(x-l(x),\infty)$
, we split (51) into three parts, respectively. Consequently, combining with Lemma 4.1 can prove (51). Likewise, (52) can be obtained. Thus, for every
$\varepsilon_2,\varepsilon_3{\,\gt\,}0$
(irrespective of
$s_1$
,
$u_1$
,
$s_{n+1}$
, and
$u_{n+1}$
) and all
$x{\,\gt\,}x_0$
, it holds uniformly for
$s_1,u_1,s_{n+1},u_{n+1}\in \Lambda_T$
and
$v,w\in[a,b]$
that
\begin{align}&\textrm{P}\!\left( vX_1+ vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,} x\big|\theta_{1}= s_1,\theta_{n+1}= s_{n+1},D_{n+1}= u_{n+1}\right)\nonumber\\[5pt] &\quad\le (1+\varepsilon_2)\!\left(\textrm{P}\!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}=s_1\right)+\textrm{P}\!\left(vX_{n+1}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1}\right)\right.\nonumber\\[5pt] &\quad\quad\!\left.+\textrm{P}\!\left(wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\right)\end{align}
and
\begin{align}&\textrm{P}\Big(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}+ vX_{n+1}+ wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\nonumber\\&{\,\gt\,} x\big|\theta_{1}= s_1,D_1= u_1,\theta_{n+1}= s_{n+1},D_{n+1}= u_{n+1}\Big)\nonumber\\[5pt] &\quad\le (1+\varepsilon_3)\!\left(\textrm{P}\!\left(wY_1\mathbb I_{\!\left\{\theta_1+D_1\le t\right\}}{\,\gt\,}x\big|\theta_{1}=s_1,D_1=u_1\right)+\textrm{P}\!\left(vX_{n+1}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1}\right)\right.\nonumber\\[5pt] &\quad\quad\!\left.+\textrm{P}\!\left(wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}=s_{n+1},D_{n+1}=u_{n+1}\right)\right).\end{align}
Substituting (47), (53), and (54) into (50) derives that
\begin{align}&\textrm{P}\Bigg(\sum\limits_{i=1}^{n+1} vX_{i} + wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}\nonumber\\[5pt] &\quad {\,\gt\,} x,vX_{n+1} + wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\le x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n+1\Bigg)\nonumber\\[5pt] &\quad\le \alpha_n \big[(1+\varepsilon_2)\textrm{P} \!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}= s_1\right)+(1+\varepsilon_3)\textrm{P} \!\left( wY_{1}\mathbb I_{\!\left\{\theta_{1}+D_{1}\le t\right\}}{\,\gt\,}x\big|\theta_{1}= s_1,D_1= u_1\right)\big]\nonumber\\[5pt] &\quad\quad+\alpha_n (\varepsilon_2+\varepsilon_3-2\varepsilon_1)\big[\textrm{P} \!\left(vX_{n+1}{\,\gt\,}x\big|\theta_{n+1}= s_{n+1}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P} \!\left( wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}{\,\gt\,}x\big|\theta_{n+1}= s_{n+1},D_{n+1}= u_{n+1}\right)\big]\end{align}
holds uniformly for
$s_{i},u_{i}\in\Lambda_T$
,
$1\le i\le n+1$
, and
$v,w\in[a,b]$
. Note that
$(X_1,Y_1,\theta_1,D_1)$
and
$(X_{n+1},Y_{n+1},\theta_{n+1},D_{n+1})$
are i.i.d., then (55) can be rewritten in the form: for every
$\varepsilon{\,\gt\,}0$
and all
$x{\,\gt\,}x_0$
,
\begin{align}&\textrm{P}\bigg(\sum\limits_{i=1}^{n+1} vX_{i}+ wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}\nonumber \\&\quad {\,\gt\,} x,vX_{n+1} + wY_{n+1}\mathbb I_{\!\left\{\theta_{n+1}+D_{n+1}\le t\right\}}\le x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n+1\bigg)\nonumber\\[5pt] &\quad\le \alpha_n(1+\varepsilon)\!\left(\textrm{P} \!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}= s_1\right)+\textrm{P} \!\left( wY_{1}\mathbb I_{\!\left\{\theta_{1}+D_{1}\le t\right\}}{\,\gt\,}x\big|\theta_{1}= s_1,D_1= u_1\right)\right)\end{align}
holds uniformly for
$s_{i},u_{i}\in\Lambda_T$
,
$1\le i\le n+1$
, and
$v,w\in[a,b]$
. Integrating with (49) we obtain
\begin{align}&\sup_{\substack{x{\,\gt\,} x_0,t\in\Lambda_T\nonumber\\[5pt] v,w\in[a,b]}}\sup_{\substack{s_{i},u_{i}\in\Lambda_T\nonumber\\[5pt] 1\le i\le n}}\frac{\textrm{P}\!\left(\sum\limits_{i=1}^{n+1} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n+1\right)}{\textrm{P} \!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}= s_1\right)+\textrm{P} \!\left( wY_{1}\mathbb I_{\!\left\{\theta_{1}+D_{1}\le t\right\}}{\,\gt\,}x\big|\theta_{1}= s_1,D_1= u_1\right)}\nonumber\\[5pt] &\quad \le (1+\varepsilon)\alpha_n+1.\end{align}
When
$x\le x_0$
, in light of inequality (46), we have
\begin{align}& \sup_{\substack{x\le x_0,t\in\Lambda_T\\ v,w\in[a,b]}} \sup_{\substack{s_{i},u_{i}\in\Lambda_T\\ 1\le i\le n+1}}\frac{\textrm{P}\!\left(\sum\limits_{i=1}^{n+1} vX_{i}+wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n+1\right)}{\textrm{P} \!\left(vX_{1}{\,\gt\,}x\big|\theta_{1}= s_1\right)+\textrm{P} \!\left( wY_{1}\mathbb I_{\!\left\{\theta_{1}+D_{1}\le t\right\}}{\,\gt\,}x\big|\theta_{1}= s_1,D_1= u_1\right)}\nonumber\\ &\quad\le \sup_{\substack{x\le x_0,t\in\Lambda_T\\ v,w\in[a,b]}} \sup_{\substack{s_{i},u_{i}\in\Lambda_T\\ 1\le i\le n+1}}\frac{1}{\frac{1}{2}\overline F_1\!\left(x_0/a\right)h(s_1)}\nonumber\\[5pt] &\quad=\mathop{\sup}\limits_{s_1\in\Lambda_T}2{\!\left(\overline F_1\!\left(x_0/a\right)h(s_1)\right)}^{-1}\nonumber\\[5pt] &\quad\le 2{\!\left(\overline F_1\!\left(x_0/a\right)\right)}^{-1}{\!\left(\mathop{\inf}\limits_{s_1\in\Lambda_T}h(s_1)\right)}^{-1}.\end{align}
Therefore, due to Assumption 2.1, there is some
$0 {\,\lt\,} A {\,\lt\,} \infty$
such that
\begin{align} \sup_{\substack{x\le x_0,t\in\Lambda_T\\[5pt] v,w\in[a,b]}} \sup_{\substack{s_{i},u_{i}\in\Lambda_T\\[5pt] 1\le i\le n+1}}\frac{\textrm{P}\!\left(\sum\limits_{i=1}^{n+1} vX_{i}+ wY_{i}\mathbb I_{\!\left\{\theta_{i}+D_{i}\le t\right\}}{\,\gt\,} x\bigg|\theta_{i}=s_{i},D_{i}=u_{i},1\le i\le n+1\right)}{\textrm{P} \!\left(vX_{1}{\,\gt\,} x\big|\theta_{1}= s_1\right)+\textrm{P} \!\left( wY_{1}\mathbb I_{\!\left\{\theta_{1}+D_{1}\le t\right\}}{\,\gt\,} x\big|\theta_{1}= s_1,D_1= u_1\right)}\le A.\end{align}
It follows from (57) and (59) that
$\alpha_{n+1}\le(1+\varepsilon)\alpha_n+1+A$
. Mimicking the proof above, we get
$\alpha_1\le1+A$
. The recursive inequality with initial value
$\alpha_1\le 1+A$
completes the proof of (45).
Next we take
$k=1$
for example to show the unidimensional form of (44), that is, for every given
$\varepsilon_1{\,\gt\,}0$
, there exists a constant
$K_1=K_1(\varepsilon_1,T)$
such that
\begin{align}&\textrm{P}\!\left(\sum\limits_{i=1}^{n} X_{1i}\textrm{e}^{-R_1(\theta_{11})}+ Y_{1i}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,} x\Bigg|\theta_{1i}= s_{1i},D_{1i}= u_{1i},1\le i \le n \right)\nonumber\\[5pt] &\quad\le K_1{(1+\varepsilon_1)}^{n}\bigg(\textrm{P} \!\left(X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)\nonumber\\[5pt] &\quad\quad+\textrm{P} \!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x\big|\theta_{11}=s_{11},D_{11}=u_{11}\right)\bigg)\end{align}
holds uniformly for
$n=1,2,\ldots$
,
$x{\,\gt\,}0$
, and
$s_{i},u_{i},t\in\Lambda_T$
,
$1\le i\le n$
. Recalling that
$\inf_{t\in\Lambda_T}\textrm{e}^{-R_1(t)}{\,\gt\,}0$
a.s., we can find a small constant
$\delta{\,\gt\,}0$
such that
$\textrm{P}\!\left(\inf_{t\in\Lambda_T}\textrm{e}^{-R_1(t)}{\,\gt\,}\delta\right){\,\gt\,}0$
. We split the left-hand probability of (60) into four parts
$J_1(x,t,\delta)$
,
$J_2(x,t,\delta)$
,
$J_3(x,t,\delta)$
, and
$J_4(x,t,\delta)$
based on its intersections with
$A_1 = \!\left\{\textrm{e}^{-R_1(s_{11})}{\,\gt\,}\delta,\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\right\}$
,
$A_2 = \!\left\{\textrm{e}^{-R_1(s_{11})}\le\delta,\textrm{e}^{-R_1(s_{11}+u_{11})}\le\delta\right\}$
,
$A_3 = \!\left\{\textrm{e}^{-R_1(s_{11})}{\,\gt\,}\delta,\textrm{e}^{-R_1(s_{11}+u_{11})}\le\delta\right\}$
, and
$A_4 = \!\left\{\textrm{e}^{-R_1(s_{11})}\le\delta,\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\right\}$
. For
$J_1(x,t,\delta)$
, using (45) in the third step derives that
\begin{align}J_1(x,t,\delta)&=\textrm{P}\!\left(\sum\limits_{i=1}^{n} X_{1i}\textrm{e}^{-R_1(\theta_{11})}+Y_{1i}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,}x,\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}\delta,\right.\nonumber\\[5pt] &\!\left.\times\textrm{e}^{-R_1(\theta_{11}+D_{11})}{\,\gt\,}\delta\big|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n\right)\nonumber\\[5pt] &=\int_\delta^1\int_\delta^1 \textrm{P}\!\left(\sum\limits_{i=1}^{n} vX_{1i}+ wY_{1i}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,} x\Bigg|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n\right)\nonumber\\[5pt] &\times\textrm{P}\!\left(\textrm{e}^{-R_1(s_{11})}\in\textrm{d} v,\textrm{e}^{-R_1(s_{11}+u_{11})}\in\textrm{d} w\right)\nonumber\\[5pt] &\le M{(1+\varepsilon_1)}^n\nonumber\\[5pt] &\int_\delta^1\int_\delta^1 \!\left(\textrm{P} \!\left( vX_{11}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)+ \textrm{P} \!\left( wY_{11}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\right)\nonumber\\[5pt] &\times\textrm{P}\!\left(\textrm{e}^{-R_1(s_{11})}\in\textrm{d} v,\textrm{e}^{-R_1(s_{11}+u_{11})}\in\textrm{d} w\right)\nonumber\\[5pt] &\le M{(1+\varepsilon_1)}^n\!\left(\textrm{P} \!\left(X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x\big|\theta_{11}= s_{11}\right)\right.\nonumber\\[5pt] &\!\left.+\textrm{P} \!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\right)\end{align}
holds uniformly for
$n=1,2,\ldots$
,
$x{\,\gt\,}0$
, and
$s_{1i},u_{1i}\in\Lambda_T$
,
$1\le i\le n$
. For
$J_2(x,t,\delta)$
, using the independence of
$\!\left\{R_1(t),t\ge0\right\}$
and
$\!\left(X_1,Y_1,\theta_1,D_1\right)$
in the fourth step we obtain
\begin{align}J_2(x,t,\delta)&=\textrm{P}\!\left(\sum\limits_{i=1}^{n} X_{1i}\textrm{e}^{-R_1(\theta_{11})}+Y_{1i}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,}x,\textrm{e}^{-R_1(\theta_{11})}\le\delta,\right.\nonumber\\[2pt] &\!\left.\times\textrm{e}^{-R_1(\theta_{11}+D_{11})}\le\delta\big|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n\right)\nonumber\\[2pt] &\le\textrm{P}\!\left(\sum\limits_{i=1}^{n} \delta X_{1i}+\delta Y_{1i}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n\right)\nonumber\\[2pt] &\le M {(1+\varepsilon_1)}^n \!\left(\textrm{P} \!\left(\delta X_{11}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)+\textrm{P} \!\left(\delta Y_{11}\mathbb I_{\!\left\{\theta_{11}+ D_{11}\le t\right\}}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\!\right)\nonumber\\[2pt] &\le M{(1+\varepsilon_1)}^n \frac{\textrm{P} \!\left(\delta X_{11}{\,\gt\,}x,\textrm{e}^{-R_1(s_{11})}{\,\gt\,}\delta\big|\theta_{11}= s_{11}\right)}{\textrm{P} \!\left(\textrm{e}^{-R_1(s_{11})}{\,\gt\,}\delta\right)}\nonumber\\[2pt] &+M{(1+\varepsilon_1)}^n\frac{\textrm{P} \!\left(\delta Y_{11}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,} x,\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)}{\textrm{P} \!\left(\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\right)}\nonumber\\[2pt] &\le\frac{M{(1+\varepsilon_1)}^n}{\textrm{P} \!\left(\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\right)}\!\left(\textrm{P} \!\left(X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x\big|\theta_{11}= s_{11}\right)\right.\nonumber\\[2pt] &\!\left.+\textrm{P} \!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\right).\end{align}
Similarly, we have
\begin{align} J_3(x,t,\delta)&=\textrm{P}\bigg(\sum\limits_{i=1}^{n} X_{1i}\textrm{e}^{-R_1(\theta_{11})}+Y_{1i}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}{\,\gt\,}x,\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}\delta, \nonumber\\[2pt] & \times\textrm{e}^{-R_1(\theta_{11}+D_{11})}\le\delta\Bigg|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n\Bigg)\nonumber\\[2pt] &\le\textrm{P}\Bigg(\sum\limits_{i=1}^{n} X_{1i}\textrm{e}^{-R_1(\theta_{11})}+\delta Y_{1i}\mathbb I_{\!\left\{\theta_{1i}+D_{1i}\le t\right\}}\nonumber\\ &\quad {\,\gt\,} x,\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}\delta\Bigg|\theta_{1i}= s_{1i},D_{1i}= u_{1i},1\le i \le n \Bigg)\nonumber\\[2pt] &\le M {(1+\varepsilon_1)}^n \!\Big(\textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,} x\big|\theta_{11}= s_{11}\right)\nonumber \\ &\quad +\textrm{P} \!\left( \delta Y_{11}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right)\Big)\nonumber\\[2pt] &\le M{(1+\varepsilon_1)}^n\textrm{P} \!\left(X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x\big|\theta_{11}= s_{11}\right)\nonumber\\[2pt] &+\frac{M{(1+\varepsilon_1)}^n}{\textrm{P} \!\left(\textrm{e}^{-R_1(s_{11}+u_{11})}{\,\gt\,}\delta\right)}\textrm{P} \!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x\big|\theta_{11}= s_{11},D_{11}= u_{11}\right),\end{align}
where the last step is the same as
$J_2(x,t,\delta)$
. Finally, since
$\textrm{e}^{-R_1(s_{11}+u_{11})}\le \textrm{e}^{-R_1(s_{11})}$
a.s., we have
\begin{align}J_4(x,t,\delta)=0.\end{align}
Combining (61)–(64) completes the proof of (60), where
$K_1=M\vee\frac{M}{\textrm{P}\!\left(\inf_{t\in\Lambda_T}\textrm{e}^{-R_1(t)}{\,\gt\,}\delta\right)}$
.
Now we are ready to prove (44). For the convenience of our presentation, we write
\begin{align*}W_{ki}=X_{ki}\textrm{e}^{-R_k(\theta_{k1})}+Y_{ki}\textrm{e}^{-R_k(\theta_{k1}+D_{k1})}\mathbb I_{\{\theta_{ki}+D_{ki}\le t\}},\quad k=1,2.\end{align*}
By conditioning on
$\sum\limits_{i=1}^{n_2} W_{2i}$
and (60), we get that for every
$n_1,n_2{\,\gt\,}0$
and
$x{\,\gt\,}0$
,
\begin{align}&\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} W_{ki}{\,\gt\,}x\Bigg|\theta_{ki}=s_{ki},D_{ki}=u_{ki},1\le i\le n_k,k=1,2\right)\nonumber \\[2pt] &=\int_0^x \textrm{P}\!\left(\sum\limits_{i=1}^{n_1} W_{1i}{\,\gt\,}x-y\Bigg|\theta_{1i}=s_{1i},D_{1i}=u_{1i},1\le i\le n_1\right)\nonumber\\[2pt] &\times\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}\in\mathrm d y\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &+\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}{\,\gt\,}x\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &\le K_1{(1+\varepsilon_1)}^{n_1}\int_0^x \!\left(\textrm{P}\!\left(X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x-y\big|\theta_{11}=s_{11}\right)\right.\nonumber\\[2pt] &\!\left.+\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x-y\big|\theta_{11}=s_{11},D_{11}=u_{11}\right)\right)\nonumber\\[2pt] &\times\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}\in\mathrm d y\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &+\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}{\,\gt\,}x\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &=K_1{(1+\varepsilon_1)}^{n_1}\Bigg[\textrm{P}\Bigg( X_{11}\textrm{e}^{-R_1(\theta_{11})}+\sum\limits_{i=1}^{n_2} W_{2i}\nonumber \\ &{\,\gt\,}x,\sum\limits_{i=1}^{n_2} W_{2i}\le x\Bigg|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\Bigg)\nonumber\\[2pt] &+\textrm{P} \Bigg( Y_{11} \textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}+\sum\limits_{i=1}^{n_2} W_{2i}\nonumber\\ &{\,\gt\,} x,\sum\limits_{i=1}^{n_2} W_{2i}\le x\Bigg|\theta_{11}= s_{11},D_{11}= u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i \le n_2\Bigg)\Bigg]\nonumber\\[2pt] &+\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}{\,\gt\,}x\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &=K_1{(1+\varepsilon_1)}^{n_1}\Bigg[\textrm{P}\Bigg( X_{11}\textrm{e}^{-R_1(\theta_{11})}+\sum\limits_{i=1}^{n_2} W_{2i}\nonumber \\ &{\,\gt\,}x\Bigg|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\Bigg)\nonumber\\[2pt] &+\textrm{P}\Bigg(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}\nonumber\\ &+\sum\limits_{i=1}^{n_2} W_{2i}{\,\gt\,}x\Bigg|\theta_{11}=s_{11},D_{11}=u_{11},\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\Bigg)\Bigg]\nonumber\\[2pt] &+(1-2K_1{(1+\varepsilon_1)}^{n_1})\textrm{P}\!\left(\sum\limits_{i=1}^{n_2} W_{2i}{\,\gt\,}x\Bigg|\theta_{2i}=s_{2i},D_{2i}=u_{2i},1\le i\le n_2\right)\nonumber\\[2pt] &\le K_1K_2{(1+\varepsilon_1)}^{n_1}{(1+\varepsilon_2)}^{n_2}\nonumber\\[2pt] &\times\!\left(\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,}x\Bigg|\theta_{11}=s_{11},\theta_{21}=s_{21}\right)\right.\nonumber\\[2pt] &+\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+ Y_{21}\textrm{e}^{-R_2(\theta_{21}+D_{21})}\mathbb I_{\!\left\{\theta_{21}+D_{21}\le t\right\}}{\,\gt\,} x\Bigg|\theta_{11}= s_{11},\theta_{21}= s_{21},D_{21}= u_{21}\right)\nonumber\\[2pt] &+\textrm{P}\!\left( Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}+ X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,} x\Bigg|\theta_{11}= s_{11},D_{11}= u_{11},\theta_{21}= s_{21}\right)\nonumber\\[2pt] &\!\left.+\textrm{P}\!\left(\sum\limits_{k=1}^2Y_{k1}\textrm{e}^{-R_k(\theta_{k1}+D_{k1})}\mathbb I_{\!\left\{\theta_{k1}+D_{k1}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2\right)\right)\end{align}
holds uniformly for
$s_{ki},u_{ki}\in\Lambda_T$
,
$1\le i\le n_k$
,
$k=1,2$
, where the last step is obtained by conditioning on
$X_{11}\textrm{e}^{-R_1(\theta_{11})}$
and
$Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\{\theta_{11}+D_{11}\le t\}}$
and applying (60) to
$\sum\limits_{i=1}^{n_2}W_{2i}$
. Take
$K=K_1K_2$
and
$\varepsilon=\varepsilon_1\vee\varepsilon_2$
. This completes the proof.
Next, we present a lemma that plays a key role in the proof of our main result, which describes the tail probability of the discounted aggregate claims from two business lines up to time t.
Lemma 4.4. Consider the bidimensional risk model (1). Under the conditions of Theorem 2.1, it holds uniformly for
$t\in\Lambda_T$
that
\begin{align}&\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{N_k(t)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x\right)\nonumber\\[2pt] &\quad\sim \sum\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k \!\left(x\textrm{e}^{R_k(s)}\right) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k \!\left(x\textrm{e}^{-R_k(s+u)}\right) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right),\end{align}
where
${\tilde\lambda}_k(t)=\int_{0-}^t (1+\lambda_k(t-s))h_k(s)H_{\theta_k}(\mathrm d s)$
and
${\hat\lambda}_k(s,u)=\int_{0-}^s(1+\lambda_k(s-v))\varphi_k(v,u)$
$H_{\theta_k}(\mathrm d v)$
.
Proof. For
$k=1,2$
, we write
\begin{align*}\gamma_k(x,t)\;:\!=\;\int_{0-}^t {\overline F}_k (x\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k (x\textrm{e}^{R_k(s+u)}) {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\end{align*}
and
$\gamma(x,t)=\gamma_1(x,t)+\gamma_2(x,t)$
. For some large integer
$m\ge1$
, we split the left side of (66) as follows:
\begin{align}&\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{N_k(t)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x\right)\nonumber\\[5pt] &\quad=\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=1}^\infty\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\tau_{ki})} + Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}} {\,\gt\,} x,N_1(t)= n_1,N_2(t)= n_2\right)\nonumber\\[5pt] &\quad=\!\left(\sum\limits_{n_1=1}^m\sum\limits_{n_2=1}^m+\sum\limits_{n_1=1}^m\sum\limits_{n_2=m+1}^\infty+\sum\limits_{n_1=m+1}^\infty\sum\limits_{n_2=1}^\infty\right)\nonumber\\[5pt] &\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+ Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,} x,N_1(t)= n_1,N_2(t)= n_2\right)\nonumber\\[5pt] &\quad\;:\!=\; Q_1(x,t)+Q_2(x,t)+Q_3(x,t).\end{align}
We use the notation
$\Gamma_t=\!\left\{s_i\le t,u_i\le t:\sum\limits_{i=1}^n s_i\le t\right\}$
. Recall that
$t_{kn}=\sum\limits_{i=1}^n s_{ki},n\in\mathbb N_+$
. For each fixed
$1\le n_1,n_2\le m$
, using Lemma 4.2 in the second step yields that
\begin{align}Q_1(x,t)&=\sum\limits_{n_1=1}^m \sum\limits_{n_2=1}^m\int_{\Gamma_t} \int_{\Gamma_t}\textrm{P} \!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} Z_{ki}{\,\gt\,} x\Bigg|\theta_{ki}= s_{ki}, D_{ki}=u_{ki},1\le i\le n_k,k =1,2\right)\nonumber\\[5pt] &\times\prod_{k=1}^2{\overline H}_{\theta_k}(t-t_{kn_k})\prod_{k=1}^2\prod_{i=1}^{n_k}H_{\theta_k}(\mathrm d s_i)H_{D_k}(\mathrm d u_i)\nonumber\\[5pt] &\sim \sum\limits_{n_1=1}^m \sum\limits_{n_2=1}^m \sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k}\int_{\Gamma_t} \int_{\Gamma_t}\!\left(\textrm{P} \!\left( X_{ki}\textrm{e}^{-R_k(t_{ki})}{\,\gt\,}x\big|\theta_{ki}=s_{ki}\right)\right.\nonumber\\[5pt] &\!\left.+\textrm{P}\!\left(Y_{ki}\textrm{e}^{-R_1(t_{ki}+u_{ki})}\mathbb I_{\!\left\{t_{ki}+u_{ki}\le t\right\}}{\,\gt\,}x\big|\theta_{ki}=s_{ki},D_{ki}=u_{ki}\right)\right)\nonumber\\[5pt] &\times\prod_{k=1}^2{\overline H}_{\theta_k}(t-t_{kn_k})\prod_{k=1}^2\prod_{i=1}^{n_k}H_{\theta_k}(\mathrm d s_i)H_{D_k}(\mathrm d u_i)\nonumber\\[5pt] &=\sum\limits_{n_1=1}^m \sum\limits_{n_2=1}^m\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k}\!\left(\textrm{P}\!\left(X_{ki}\textrm{e}^{-R_k(\tau_{ki})}{\,\gt\,}x,N_1(t)=n_1,N_2(t)=n_2\right)\right.\nonumber\\[5pt] &\!\left.+\textrm{P}\!\left(Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{t_{ki}+u_{ki}\le t\right\}}{\,\gt\,}x,N_1(t)=n_1,N_2(t)=n_2\right)\right)\nonumber\\[5pt] &=\!\left(\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=1}^\infty-\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=m+1}^\infty-\sum\limits_{n_1=m+1}^\infty\sum\limits_{n_2=1}^m\right)\nonumber\\[5pt] &\times\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k}\!\left(\textrm{P}\!\left(X_{ki}\textrm{e}^{-R_k(\tau_{ki})}{\,\gt\,}x,N_1(t)=n_1,N_2(t)=n_2\right)\right.\nonumber\\[5pt] &\!\left.+\textrm{P}\!\left(Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{t_{ki}+u_{ki}\le t\right\}}{\,\gt\,}x,N_1(t)=n_1,N_2(t)=n_2\right)\right)\nonumber\\[5pt] \;:\!=\;&Q_{11}-Q_{12}-Q_{13}\nonumber\\[5pt] &=(Q_{111}+Q_{112}+Q_{113}+Q_{114})-(Q_{121}+Q_{122}+Q_{123}+Q_{124})-Q_{13}.\end{align}
First, we analyze
$Q_{111}$
. Interchanging the order of sums in
$Q_{111}$
and conditioning on
$\tau_{1,i-1}$
and
$\theta_{1i}$
lead to
\begin{align}Q_{111}&=\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=1}^\infty\sum\limits_{i=1}^{n_1}\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(\tau_{1i})}{\,\gt\,}x,N_1(t)=n_1\right)\textrm{P}(N_2(t)=n_2)\nonumber\\[5pt] &=\sum\limits_{i=1}^\infty\sum\limits_{n_1=i}^{\infty}\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(\tau_{1i})}{\,\gt\,}x,N_1(t)=n_1\right)\nonumber\\[5pt] &=\sum\limits_{i=1}^\infty\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(\tau_{1i})}{\,\gt\,}x,\tau_{1i}\le t\right)\nonumber\\[5pt] &\sim\sum\limits_{i=1}^\infty\int_{0-}^t\int_{0-}^{t-u}\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(u+v)}{\,\gt\,}x\right)\textrm{P}\!\left(\tau_{1,i-1}\in\mathrm d v\right)h_1(u)H_{\theta_1}(\textrm{d} u)\nonumber\\[5pt] &=\int_{0-}^t\textrm{P}\!\left(X_{1}\textrm{e}^{-R_1(s)}{\,\gt\,}x\right){\tilde\lambda}_1(\textrm{d} s),\end{align}
where the last step follows from (4.18) of [Reference Li, Tang and Wu20]. Symmetrically, it can be shown that
\begin{align}Q_{112}&=\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=1}^\infty\sum\limits_{i=1}^{n_1}\textrm{P}\!\left(Y_{1i}\textrm{e}^{-R_1(\tau_{1i}+D_{1i})}\mathbb I_{\!\left\{t_{1i}+u_{1i}\le t\right\}}{\,\gt\,}x,N_1(t)=n_1\right)\textrm{P}(N_2(t)=n_2)\nonumber\\[5pt] &=\sum\limits_{i=1}^\infty\textrm{P}\!\left(Y_{1i}\textrm{e}^{-R_1(\tau_{1i}+D_{1i})}\mathbb I_{\!\left\{t_{1i}+u_{1i}\le t\right\}}{\,\gt\,}x,\tau_{1i}\le t\right)\nonumber\\[5pt] &\sim\sum\limits_{i=1}^\infty\int_{0-}^t \int_{0-}^{t-u}\int_{0-}^{t-u-s}\textrm{P} \!\left( Y_{1i}\textrm{e}^{-R_1(u+v+s)}{\,\gt\,} x \right)\textrm{P} \!\left(\tau_{1,i-1}\in\mathrm d v \right)\varphi_1(s,u)H_{\theta_1}(\textrm{d} s)H_{D_1}(\textrm{d} u)\nonumber\\[5pt] &=\int_{0-}^t \int_{0-}^{t-u}\bigg(\textrm{P}\!\left(Y_{1}\textrm{e}^{-R_1(u+s)}{\,\gt\,}x\right)\nonumber \\ &\quad +\int_{0-}^{t-u-s}\textrm{P}\!\left(Y_{1}\textrm{e}^{-R_1(u+v+s)}{\,\gt\,} x\right)\lambda_1(\textrm{d} v)\bigg) \varphi_1(s,u)H_{\theta_1}(\textrm{d} s)H_{D_1}(\textrm{d} u)\nonumber\\[5pt] &=\int_{0-}^t\int_{0-}^{t-u} \textrm{P}\!\left(Y_{1}\textrm{e}^{-R_1(u+s)}{\,\gt\,}x\right) {\hat\lambda}_1(\mathrm d s,u) H_{D_1}(\mathrm d u),\end{align}
\begin{align}Q_{113}\sim\int_{0-}^t\textrm{P}\!\left(X_{2}\textrm{e}^{-R_2(s)}{\,\gt\,}x\right){\tilde\lambda}_2(\textrm{d} s)\end{align}
and
\begin{align}Q_{114}\sim\int_{0-}^t\int_{0-}^{t-u}\textrm{P}\!\left(Y_{2}\textrm{e}^{-R_2(u+s)}{\,\gt\,}x\right) {\hat\lambda}_2(\mathrm d s,u) H_{D_2}(\mathrm d u).\end{align}
Thus, we have
$Q_{11}\sim\gamma(x,t)$
. For every
$\delta_1{\,\gt\,}0$
, we can choose some large positive integer m such that
$\textrm{P}(N_2(t)\ge m)\le\delta_1$
uniformly for
$t\in\Lambda_T$
. Then we have
\begin{align}Q_{121}&=\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=m+1}^\infty\sum\limits_{i=1}^{n_1}\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(\tau_{1i})}{\,\gt\,}x,N_1(t)=n_1\right)\textrm{P}(N_2(t)=n_2)\nonumber\\[5pt] &=\sum\limits_{i=1}^\infty\textrm{P}\!\left(X_{1i}\textrm{e}^{-R_1(\tau_{1i})}{\,\gt\,}x,\tau_{1i}\le t\right)\textrm{P}(N_2(t)\ge m+1)\nonumber\\[5pt] &\le\delta_1\!\left(\int_{0-}^t\textrm{P}\!\left(X_{1}\textrm{e}^{-R_1(s)}{\,\gt\,}x\right) {\tilde\lambda}_1(\mathrm d s)\right)\end{align}
and
\begin{align}Q_{122}\le\delta_1\!\left(\int_{0-}^t\int_{0-}^{t-u} \textrm{P}\!\left(Y_{1}\textrm{e}^{-R_1(u+s)}{\,\gt\,}x\right) {\hat\lambda}_1(\mathrm d s,u) H_{D_1}(\mathrm d u)\right).\end{align}
As for
$Q_{123}$
and
$Q_{124}$
, note that
$\textrm{E}(N_2(t)+1)\mathbb I_{\!\left\{N_2(t)\ge m\right\}}\to0$
as
$m\to\infty$
. For every
$\delta_2{\,\gt\,}0$
, we can find some large positive integer m such that, uniformly for
$t\in\Lambda_T$
,
\begin{align}Q_{123}&=\sum\limits_{n_1=1}^\infty\sum\limits_{n_2=m+1}^\infty\sum\limits_{i=1}^{n_2}\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(\tau_{2i})}{\,\gt\,}x,N_2(t)=n_2\right)\textrm{P}(N_1(t)=n_1)\nonumber\\[5pt] &=\sum\limits_{n_2=m+1}^\infty\sum\limits_{i=1}^{n_2}\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(\tau_{2i})}{\,\gt\,}x,N_2(t)=n_2\right)\nonumber\\[5pt] &\le\sum\limits_{n_2=m+1}^\infty\sum\limits_{i=1}^{n_2}\int_{0-}^t\textrm{P}\!\left(X_{2i}\textrm{e}^{-R_2(s)}{\,\gt\,}x\big|\theta_{2i}=s\right)\textrm{P}\!\left(N_2(t-s)=n_2-1\right)H_{\theta_2}(\mathrm d s)\nonumber\\[5pt] &\sim\sum\limits_{n_2=m+1}^\infty n_2\int_{0-}^t\textrm{P}\!\left(X_{2}\textrm{e}^{-R_2(s)}{\,\gt\,}x\right)\textrm{P}\!\left(N_2(t-s)=n_2-1\right)h_2(s)H_{\theta_2}(\mathrm d s)\nonumber\\[5pt] &\le \textrm{E}(N_2(t)+1)\mathbb I_{\!\left\{N_2(t)\ge m\right\}}\int_{0-}^t\textrm{P}\!\left(X_{2}\textrm{e}^{-R_2(s)}{\,\gt\,}x\right) {\tilde\lambda}_2(\mathrm d s)\nonumber\\[5pt] &\le\delta_2\!\left(\int_{0-}^t\textrm{P}\!\left(X_{2}\textrm{e}^{-R_2(s)}{\,\gt\,}x\right) {\tilde\lambda}_2(\mathrm d s)\right)\end{align}
and
\begin{align}Q_{124}\lesssim\delta_2\!\left(\int_{0-}^t\int_{0-}^{t-u} \textrm{P}\!\left(Y_{2}\textrm{e}^{-R_2(u+s)}{\,\gt\,}x\right) {\hat\lambda}_2(\mathrm d s,u) H_{D_2}(\mathrm d u)\right).\end{align}
Combining (73)–(76) and noting the arbitrariness of
$\delta_1$
and
$\delta_2$
lead that
$Q_{12}=o(Q_{11})$
holds uniformly for
$t\in\Lambda_T$
. Similarly we get
$Q_{13}=o(Q_{11})$
. Thus,
$Q_1\sim Q_{11}\sim\gamma(x,t)$
. Next we focus on
$Q_2$
. Applying Lemma 4.3 in the third step and recalling the independence of
$\!\left(X_k,\theta_k\right)$
and
$D_k$
, we obtain, uniformly for
$t\in\Lambda_T$
,
\begin{align}Q_2(x,t)&=\sum\limits_{n_1=1}^m\sum\limits_{n_2=m+1}^\infty\nonumber\\[5pt] &\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x,N_1(t)=n_1,N_2(t)=n_2\right)\nonumber\\[5pt] &\le\sum\limits_{n_1=1}^m \sum\limits_{n_2=m+1}^\infty\int_{\Gamma_t}\int_{\Gamma_t}\textrm{P} \Bigg(\sum\limits_{k=1}^2\sum\limits_{i=1}^{n_k} X_{ki}\textrm{e}^{-R_k(\theta_{k1})}+ Y_{ki}\textrm{e}^{-R_k(\theta_{k1}+ D_{k1})}\mathbb I_{\!\left\{\theta_{ki}+ D_{ki}\le t \right\}}\nonumber \\ &{\,\gt\,} x\Bigg|\theta_{ki}= s_{ki},D_{ki}= u_{ki},1\le i \le n_k,k =1,2\Bigg)\nonumber\\[5pt] &\times\overline H_{\theta_1}(t-t_{1n_1})\overline H_{\theta_2}(t-t_{2n_2})\prod_{k=1}^2\prod_{i=1}^{n_k}H_{\theta_k}(\mathrm d s_{ki})H_{D_k}(\mathrm d u_{ki})\nonumber\\[5pt] &\le\sum\limits_{n_1=1}^m\sum\limits_{n_2=m+1}^\infty K{(1+\varepsilon)}^{n_1+n_2}\nonumber\\[5pt] &\times \int_{\Gamma_t}\int_{\Gamma_t}\bigg(\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,}x\big|\theta_{11}=s_{11},\theta_{21}=s_{21}\right)\nonumber\\[5pt] &+\textrm{P} \!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}+ Y_{21}\textrm{e}^{-R_2(\theta_{21}+D_{21})}\mathbb I_{\!\left\{\theta_{21}+D_{21}\le t\right\}}{\,\gt\,} x\big|\theta_{11}= s_{11},\theta_{21}= s_{21},D_{21}= u_{21}\right)\nonumber\\[5pt] &+\textrm{P} \!\left( Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}+ X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,} x\big|\theta_{11}= s_{11},D_{11}= u_{11},\theta_{21}= s_{21}\right)\nonumber\\[5pt] &\!\left.+\textrm{P}\!\left(\sum\limits_{k=1}^2Y_{k1}\textrm{e}^{-R_k(\theta_{k1}+D_{k1})}\mathbb I_{\!\left\{\theta_{k1}+D_{k1}\le t\right\}}{\,\gt\,}x\Bigg|\theta_{k1}=s_{k1},D_{k1}=u_{k1},k=1,2\right)\right)\nonumber\\[5pt] &\times\overline H_{\theta_1}(t-t_{1n_1})\overline H_{\theta_2}(t-t_{2n_2})\prod_{k=1}^2\prod_{i=1}^{n_k}H_{\theta_k}(\mathrm d s_{ki})H_{D_k}(\mathrm d u_{ki})\nonumber\\[5pt] \lesssim&\sum\limits_{n_1=1}^m\sum\limits_{n_2=m+1}^\infty 2K{(1+\varepsilon)}^{n_1+n_2}\textrm{P}(N_2(t)=n_2)\nonumber\\[5pt] &\int_{0-}^t\int_{0-}^t\bigg(\textrm{P}\!\left( X_{11}\textrm{e}^{-R_1(\theta_{11})}{\,\gt\,}x\big|\theta_{11}=s_{11}\right)\nonumber\\[5pt] &+\textrm{P}\!\left(Y_{11}\textrm{e}^{-R_1(\theta_{11}+D_{11})}\mathbb I_{\!\left\{\theta_{11}+D_{11}\le t\right\}}{\,\gt\,}x\big|\theta_{11}=s_{11},D_{11}=u_{11}\right)\bigg)\nonumber\\[5pt] &\times\textrm{P}\!\left(N_1(t-s_{11})=n_1-1\right)H_{\theta_{1}}(\mathrm d s_{11})H_{D_1}(\mathrm d u_{11})\nonumber\\[5pt] &+\sum\limits_{n_1=1}^m\sum\limits_{n_2=m+1}^\infty 2K{(1+\varepsilon)}^{n_1+n_2}\textrm{P}(N_1(t)=n_1)\nonumber\\[5pt] &\times\int_{0-}^t\int_{0-}^t\bigg(\textrm{P}\!\left( X_{21}\textrm{e}^{-R_2(\theta_{21})}{\,\gt\,}x\big|\theta_{21}=s_{21}\right)\nonumber\\[5pt] &+\textrm{P}\!\left(Y_{21}\textrm{e}^{-R_2(\theta_{21}+D_{21})}\mathbb I_{\!\left\{\theta_{21}+D_{21}\le t\right\}}{\,\gt\,}x\big|\theta_{21}=s_{21},D_{21}=u_{21}\right)\bigg)\nonumber\\[5pt] &\times\textrm{P}\!\left(N_2(t-s_{21})=n_2-1\right)H_{\theta_2}(\mathrm d s_{21})H_{D_2}(\mathrm d u_{21})\nonumber\\[5pt] &\le\sum\limits_{n_1=1}^{\infty}\sum\limits_{n_2=m+1}^\infty 2K{(1+\varepsilon)}^{n_1+n_2}\textrm{P}(N_1(t)=n_1-1)\textrm{P}(N_2(t)=n_2)\gamma_1(x,t)\nonumber\\[5pt] &+\sum\limits_{n_1=1}^{\infty}\sum\limits_{n_2=m+1}^\infty 2K{(1+\varepsilon)}^{n_1+n_2}\textrm{P}(N_1(t)=n_1)\textrm{P}(N_2(t)=n_2-1)\gamma_2(x,t)\nonumber\\[5pt] &\le2K(1+\varepsilon)\gamma_1(x,t)\textrm{E}{(1+\varepsilon)}^{N_1(t)}\textrm{E}{(1+\varepsilon)}^{N_2(t)}\mathbb I_{\!\left\{N_2(t)\ge m+1\right\}}\nonumber\\[5pt] &+2K(1+\varepsilon)\gamma_2(x,t)\textrm{E}{(1+\varepsilon)}^{N_1(t)}\textrm{E}{(1+\varepsilon)}^{N_2(t)+1}\mathbb I_{\!\left\{N_2(t)\ge m\right\}}\nonumber\\[5pt] &\le2K(1+\varepsilon)\gamma(x,t)\textrm{E}{(1+\varepsilon)}^{N_1(t)}\textrm{E}{(1+\varepsilon)}^{N_2(t)+1}\mathbb I_{\!\left\{N_2(t)\ge m\right\}},\end{align}
where the fourth step can be obtained by the same method as in the proof of (22) in Lemma 4.2 and the fifth step can be referenced from (69)–(72). By Theorem 1 of [Reference Kocetova, Leipus and Siaulys14], we may choose some
$\varepsilon$
sufficiently small such that
$\textrm{E}{(1+\varepsilon)}^{N_1(t)}{\,\lt\,}\infty$
and
$\textrm{E}{(1+\varepsilon)}^{N_2(t)+1}\mathbb I_{\!\left\{N_2(t)\ge m\right\}}\to0$
as
$m\to\infty$
. Thus,
$Q_2{\,\lt\,}\varepsilon\gamma(x,t)$
. Similarly, we have, for every
$\tilde\varepsilon{\,\gt\,}0$
,
$Q_3{\,\lt\,}\tilde\varepsilon\gamma(x,t)$
. Letting
$\varepsilon,\tilde\varepsilon\to0$
completes the proof.
Next, we show that (8) can be derived from (9) under some certain conditions.
Lemma 4.5. Suppose that the random vector
$(X,Y,\theta,D)$
satisfies Assumption 2.1 and (9) with
$\overline F(x)\asymp\overline G(x)\in\mathcal L\cap\mathcal D$
. Then there are some nonnegative Lévy processes
$\!\left\{R(t),t\ge 0\right\}$
such that (8) holds.
Proof. The proof to verify (8) follows the same approach of Lemma A.5 in [Reference Yuan and Lu32], with certain points to be further clarified. First, in the present lemma, the weights of X and Y change from measurable functions
$g_1(x), g_2(x,y)\in[a,b]$
for
$0 {\,\lt\,} a {\,\lt\,} b {\,\lt\,} \infty$
to nonincreasing stochastic processes
$0 {\,\lt\,} \textrm{e}^{-R(\theta+D)}\le \textrm{e}^{-R(\theta)}{\,\lt\,}1$
a.s. However, Lemma 4.1 ensures that the uniformity of the asymptotic formulas remain valid. Second, when dealing with
$\textrm{P}\!\left(X\textrm{e}^{-R(\theta)}{\,\gt\,}x/2\big|\theta=s\right)$
and
$\textrm{P}\!\left(Y\textrm{e}^{-R(\theta+D)}{\,\gt\,}x/2\big|\theta=s,D=u\right)$
in
$I_3$
, we use the closure property of class
$\mathcal D$
with respect to convolution product in the second step, see Proposition 5.4(i) of [Reference Leipus, Siaulys and Konstantinides16]. Thus, we have
\begin{align}\textrm{P} \!\left( X\textrm{e}^{-R(\theta)}{\,\gt\,}\frac{x}{2}\bigg|\theta = s \right)\sim\textrm{P} \!\left( X\textrm{e}^{-R(\theta)}{\,\gt\,}\frac{x}{2}\right) h(s)\asymp\textrm{P} ( X\textrm{e}^{-R(\theta)}{\,\gt\,} x) h(s)\sim\textrm{P} \!\left( X\textrm{e}^{-R(\theta)}{\,\gt\,} x\Big|\theta= s \right)\end{align}
and
\begin{align}&\textrm{P}\!\left(Y\textrm{e}^{-R(\theta+D)}{\,\gt\,}\frac{x}{2}\bigg|\theta=s,D=u\right)\sim P\!\left(Y\textrm{e}^{-R(\theta+D)}{\,\gt\,}\frac{x}{2}\right)\varphi(s,u)\nonumber\\[5pt] &\quad\asymp P\!\left(Y\textrm{e}^{-R(\theta+D)}{\,\gt\,} x\right)\varphi(s,u)\sim\textrm{P}\!\left(Y\textrm{e}^{-R(\theta+D)}{\,\gt\,} x\big|\theta=s,D=u\right).\end{align}
Consequently, we can obtain
$I_3 = o(I_1+I_2)$
. This complete the proof.
4.2. Proof of Theorem 2.1
Let
$x=x_1+x_2$
,
$\widetilde C_k(t)=\int_0^t\textrm{e}^{-R_k(s)}C_k(\mathrm d s)$
,
$k=1,2$
, and
$\widetilde C(t)=\widetilde C_1(t)+\widetilde C_2(t)$
, where
$\widetilde C_k(t)$
represents the total discounted value of premiums accumulated up to time
$t{\,\gt\,}0$
for
$k=1,2$
. We investigate
$\psi_{\rm sum}(x_1,x_2;\;t)$
first. For the asymptotic upper bound, it can be derived that
\begin{align}\psi_{\rm sum}(x_1,x_2;\;t)&=\textrm{P}\!\left(\inf\limits_{0\le s\le t}\!\left\{U_1(s)+U_2(s)\right\}{\,\lt\,}0\big|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right)\nonumber\\[5pt] &=\textrm{P}\!\left(\sup\limits_{0\le s\le t}\!\left\{\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{N_k(s)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})} + Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le s\right\}}\right) - \widetilde C(s)\right\} {\,\gt\,} x\right)\nonumber\\[5pt] &\le\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{N_k(t)}X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x\right)\nonumber\\[5pt] &\sim \gamma(x,t),\end{align}
where Lemma 4.4 is used in the final step. Since
$\widetilde C(t)$
is nonnegative and independent of the other randomness, conditioning on
$\widetilde C(T)$
and applying the dominated convergence argument and Lemma 4.4, we obtain
\begin{align}\psi_{\rm sum}(x_1,x_2;\;t)&\ge\textrm{P}\!\left(\sum\limits_{k=1}^2\sum\limits_{i=1}^{N_k(t)}X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x+\widetilde C(T)\right)\nonumber\\[5pt] &\sim \sum\limits_{k=1}^2\bigg(\int_0^t \textrm{P}\!\left(X_k\textrm{e}^{-R_k(s)}-\widetilde C(T) {\,\gt\,}x\right){\widetilde\lambda}_k(\textrm{d} s)\nonumber\\[5pt] &+\int_0^t \int_0^{t-u} \textrm{P}\!\left( Y_k\textrm{e}^{-R_k(s+u)}-\widetilde C(T){\,\gt\,}x\right){\hat\lambda}_k(\textrm{d} s,u)H_{D_k}(\textrm{d} u)\bigg)\nonumber\\[5pt] &\sim \gamma(x,t),\end{align}
where in the ultimate step we use the assumption
$\textrm{P}\!\left(\tilde C(T){\,\gt\,}x\right)=o(\overline F_k(x/a))$
for any fixed
$T{\,\gt\,}0$
and some
$a{\,\gt\,}0$
and Lemma 2.5 of [Reference Wang, Chen, Yang and Gao26]. Combining (80) and (81) gives the asymptotic formula of
$\psi_{\rm sum}(x_1,x_2;\;t)$
. Now we deal with
$\psi_{\rm or}(x_1,x_2;\;t)$
. Note that
\begin{align}\psi_{\rm or}(x_1,x_2;\;t)&=\textrm{P}\!\left(\bigcup\limits_{k=1}^2\!\left\{\inf\limits_{0\le s\le t}\textrm{e}^{-R_k(s)}U_k(s){\,\lt\,}0\right\}\Bigg|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right)\nonumber\\[2pt] &=\textrm{P}\!\left(\bigcup\limits_{k=1}^2\!\left\{\sup\limits_{0\le s\le t}\!\left(\sum\limits_{i=1}^{N_k(s)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})} + Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le s\right\}}\!\right) - \widetilde C_k(s) {\,\gt\,} x_k\!\right\}\!\right)\!.\nonumber\\[5pt] \end{align}
Notably, following the same approach of proving Lemma 4.4, for
$k=1,2$
, we have
\begin{align}&\textrm{P}\!\left(\sum\limits_{i=1}^{N_k(t)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x\right)\nonumber\\[2pt] &\quad \sim \!\left(\int_{0-}^t {\overline F}_k \!\left(x\textrm{e}^{R_k(s)}\right) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k \!\left(x\textrm{e}^{-R_k(s+u)}\right) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right),\end{align}
holds uniformly for
$t\in\Lambda_T$
. Consequently, we obtain the asymptotic upper bound of
$\psi_{\rm or}(x_1,x_2;\;t)$
by (83), that is
\begin{align}\psi_{\rm or}(x_1,x_2;\;t)&\le\textrm{P}\!\left(\bigcup\limits_{k=1}^2\!\left\{\sum\limits_{i=1}^{N_k(t)}X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x_k\right\}\right)\nonumber\\[2pt] &=\sum\limits_{k=1}^2\textrm{P}\!\left(\sum\limits_{i=1}^{N_k(t)}X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}{\,\gt\,}x_k\right)\nonumber\\[2pt] &\sim \sum\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k (x_k\textrm{e}^{R_k(s+u)}) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right),\end{align}
where the last step follows from Lemma 4.4. Conditioning on
$\widetilde C_1(t)$
and
$\widetilde C_2(t)$
in the second step and using (83) and Lemma 2.5 of [Reference Wang, Chen, Yang and Gao26], we have
\begin{align}\!\!\!\!\psi_{\rm or}(x_1,x_2;\;t) &\ge\textrm{P}\!\left(\bigcup\limits_{k=1}^2\!\left\{\sum\limits_{i=1}^{N_k(t)}X_{ki}\textrm{e}^{-R_k(\tau_{ki})}+Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le t\right\}}-\widetilde C_k(t){\,\gt\,}x_k\right\}\right)\nonumber\\[2pt] &\sim \sum\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k (x_k\textrm{e}^{R_k(s+u)}) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\nonumber\\[2pt] &-\prod\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k (x_k\textrm{e}^{R_k(s+u)}) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\nonumber\\[2pt] &\sim \sum\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k (x_k\textrm{e}^{R_k(s+u)}) {\hat\lambda}_k (\mathrm d s,u) H_{D_k}(\mathrm d u)\right),\end{align}
where the last step holds because the product term is o(1) compared with the sum term. Now we consider
$\psi_{\rm and}(x_1,x_2;\;t)$
and
$\psi_{\rm sim}(x_1,x_2;\;t)$
. Similarly, as
$\psi_{\rm or}$
, we derive
\begin{align}\psi_{\rm and}(x_1,x_2;\;t)&=\textrm{P}\!\left(\bigcap\limits_{k=1}^2\!\left\{\sup\limits_{0\le s\le t}\!\left(\sum\limits_{i=1}^{N_k(s)} X_{ki}\textrm{e}^{-R_k(\tau_{ki})} + Y_{ki}\textrm{e}^{-R_k(\tau_{ki}+D_{ki})}\mathbb I_{\!\left\{\tau_{ki}+D_{ki}\le s\right\}}\right) - \widetilde C_{k}(s) {\,\gt\,} x_k\!\right\}\!\right)\nonumber\\[5pt] &\sim \prod\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k \Big(x_k\textrm{e}^{R_k(s+u)}\Big) {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\right)\end{align}
and
\begin{align}\psi_{\rm and}(x_1,x_2;\;t)&\ge\psi_{\rm sim}(x_1,x_2,t)\ge\textrm{P}\!\left( U_1(t){\,\lt\,}0,U_2(t){\,\lt\,}0\big|{(U_1(0),U_2(0))}^\top={(x_1,x_2)}^\top\right)\nonumber\\[5pt] &\sim \prod\limits_{k=1}^2\!\left(\int_{0-}^t {\overline F}_k (x_k\textrm{e}^{R_k(s)}) {\tilde\lambda}_k(\mathrm d s)+\int_{0-}^t\int_{0-}^{t-u} {\overline G}_k \Big(x_k\textrm{e}^{R_k(s+u)}\Big) {\hat\lambda}_k(\mathrm d s,u) H_{D_k}(\mathrm d u)\right).\end{align}
Plugging (86) into (87) leads the asymptotic formulas of
$\psi_{\rm sim}(x_1,x_2;\;t)$
. This completes the proof of Theorem 2.1.
Funding information
The first author was supported by National Key R&D Program of China (No. 2023YFA1009200), National Natural Science Foundation of China (No. 12371154), the Fundamental Research Funds for the Central Universities (No. DUT23LAB307), Dalian High-level Talent Innovation Program (No. 2020RD09), and Natural Science Foundation of Inner Mongolia (No. 2025MS01025). The third author was supported by Scientific Research Fund of Liaoning Provincial Education Department (No. LJ212510173010).
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
Data
In the simulation part of this paper, we did not use empirical data. All data generated during the current study are available from the corresponding author on reasonable request.
Acknowledgements
We sincerely thank the referees for their constructive and insightful suggestions, which have enhanced the clarity and completeness of our paper significantly. We would like to clarify that the authors contributed equally.
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